Areas

#$&*

course Phy 122

1/12 4pm

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

qa areas etc

001. Areas

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Question: `q001. There are 11 questions and 7 summary questions in this assignment.

What is the area of a rectangle whose dimensions are 4 m by 3 meters.

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Your solution:

12 m^2 (4m x 3m)

confidence rating #$&*: 3

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Given Solution:

`aA 4 m by 3 m rectangle can be divided into 3 rows of 4 squares, each 1 meter on a side. This makes 3 * 4 = 12 such squares. Each 1 meter square has an area of 1 square meter, or 1 m^2. The total area of the rectangle is therefore 12 square meters, or 12 m^2.

The formula for the area of a rectangle is A = L * W, where L is the length and W the width of the rectangle. Applying this formula to the present problem we obtain area A = L * W = 4 m * 3 m = (4 * 3) ( m * m ) = 12 m^2.

Note the use of the unit m, standing for meters, in the entire calculation. Note that m * m = m^2.

FREQUENT STUDENT ERRORS

The following are the most common erroneous responses to this question:

4 * 3 = 12

4 * 3 = 12 meters

INSTRUCTOR EXPLANATION OF ERRORS

Both of these solutions do indicate that we multiply 4 by 3, as is appropriate.

However consider the following:

4 * 3 = 12.

4 * 3 does not equal 12 meters.

4 * 3 meters would equal 12 meters, as would 4 meters * 3.

However the correct result is 4 meters * 3 meters, which is not 12 meters but 12 meters^2, as shown in the given solution.

To get the area you multiply the quantities 4 meters and 3 meters, not the numbers 4 and 3. And the result is 12 meters^2, not 12 meters, and not just the number 12.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q002. What is the area of a right triangle whose legs are 4.0 meters and 3.0 meters?

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Your solution:

6m^2 (1/2 * 4m *3m)

confidence rating #$&*: 3

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Given Solution:

`aA right triangle can be joined along its hypotenuse with another identical right triangle to form a rectangle. In this case the rectangle would have dimensions 4.0 meters by 3.0 meters, and would be divided by any diagonal into two identical right triangles with legs of 4.0 meters and 3.0 meters.

The rectangle will have area A = L * W = 4.0 m * 3.0 m = 12 m^2, as explained in the preceding problem. Each of the two right triangles, since they are identical, will therefore have half this area, or 1/2 * 12 m^2 = 6.0 m^2.

The formula for the area of a right triangle with base b and altitude h is A = 1/2 * b * h.

STUDENT QUESTION

Looking at your solution I think I am a bit rusty on finding the area of triangles. Could you give me a little more details

on how you got your answer?

INSTRUCTOR RESPONSE

As explained, a right triangle is half of a rectangle.

There are two ways to put two right triangles together, joining them along the hypotenuse. One of these ways gives you a rectangle. The common hypotenuse thus forms a diagonal line across the rectangle.

The area of either triangle is half the area of this rectangle.

If this isn't clear, take a blade or a pair of scissors and cut a rectangle out of a piece of paper. Make sure the length of the rectangle is clearly greater than its width. Then cut your rectangle along a diagonal, to form two right triangles.

Now join the triangles together along the hypotenuse. They will either form a rectangle or they won't. Either way, flip one of your triangles over and again join them along the hypotenuse. You will have joined the triangles along a common hypotenuse, in two different ways. If you got a rectangle the first time, you won't have one now. And if you have a rectangle now, you didn't have one the first time.

It should be clear that the two triangles have equal areas (allowing for a little difference because we can't really cut them with complete accuracy).

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q003. What is the area of a parallelogram whose base is 5.0 meters and whose altitude is 2.0 meters?

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Your solution:

10m^2 (5m * 2m)

confidence rating #$&*:

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Given Solution:

`aA parallelogram is easily rearranged into a rectangle by 'cutting off' the protruding end, turning that portion upside down and joining it to the other end. Hopefully you are familiar with this construction. In any case the resulting rectangle has sides equal to the base and the altitude so its area is A = b * h.

The present rectangle has area A = 5.0 m * 2.0 m = 10 m^2.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q004. What is the area of a triangle whose base is 5.0 cm and whose altitude is 2.0 cm?

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Your solution:

10cm^2 (1/2 * 5cm * 2cm)

confidence rating #$&*: 3

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Given Solution:

`aIt is possible to join any triangle with an identical copy of itself to construct a parallelogram whose base and altitude are equal to the base and altitude of the triangle. The area of the parallelogram is A = b * h, so the area of each of the two identical triangles formed by 'cutting' the parallelogram about the approriate diagonal is A = 1/2 * b * h. The area of the present triangle is therefore A = 1/2 * 5.0 cm * 2.0 cm = 1/2 * 10 cm^2 = 5.0 cm^2.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q005. What is the area of a trapezoid with a width of 4.0 km and average altitude of 5.0 km?

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Your solution:

20km^2 (4km * 5km)

confidence rating #$&*: 3

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Given Solution:

`aAny trapezoid can be reconstructed to form a rectangle whose width is equal to that of the trapezoid and whose altitude is equal to the average of the two altitudes of the trapezoid. The area of the rectangle, and therefore the trapezoid, is therefore A = base * average altitude. In the present case this area is A = 4.0 km * 5.0 km = 20 km^2.

STUDENT SOLUTION ILLUSTRATING NEED TO USE UNITS IN ALL STEPS

A=Base time average altitude therefore………A=4 *5= 20 km ^2

INSTRUCTOR COMMENT

A = (4 km) * (5 km) = 20 km^2.

Use the units at every step. km * km = km^2, and this is why the answer comes out in km^2.

Try to show the units and how they work out in every step of the solution.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q006. What is the area of a trapezoid whose width is 4 cm in whose altitudes are 3.0 cm and 8.0 cm?

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Your solution:

22cm^2 (4cm * [(8cm+3cm)/2])

confidence rating #$&*: 3

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Given Solution:

`aThe area is equal to the product of the width and the average altitude. Average altitude is (3 cm + 8 cm) / 2 = 5.5 cm so the area of the trapezoid is A = 4 cm * 5.5 cm = 22 cm^2.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q007. What is the area of a circle whose radius is 3.00 cm?

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Your solution:

28.3cm^2 (pi * 3cm^2)

confidence rating #$&*: 3

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Given Solution:

`aThe area of a circle is A = pi * r^2, where r is the radius. Thus

A = pi * (3 cm)^2 = 9 pi cm^2.

Note that the units are cm^2, since the cm unit is part r, which is squared.

The expression 9 pi cm^2 is exact. Any decimal equivalent is an approximation. Using the 3-significant-figure approximation pi = 3.14 we find that the approximate area is A = 9 pi cm^2 = 9 * 3.14 cm^2 = 28.26 cm^2, which we round to 28.3 cm^2 to match the number of significant figures in the given radius.

Be careful not to confuse the formula A = pi r^2, which gives area in square units, with the formula C = 2 pi r for the circumference. The latter gives a result which is in units of radius, rather than square units. Area is measured in square units; if you get an answer which is not in square units this tips you off to the fact that you've made an error somewhere.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q008. What is the circumference of a circle whose radius is exactly 3 cm?

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Your solution:

18.8cm (2*pi*3cm)

confidence rating #$&*:

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Given Solution:

`aThe circumference of this circle is

C = 2 pi r = 2 pi * 3 cm = 6 pi cm.

This is the exact area. An approximation to 3 significant figures is 6 * 3.14 cm = 18.8 cm.

Note that circumference is measured in the same units as radius, in this case cm, and not in cm^2. If your calculation gives you cm^2 then you know you've done something wrong.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q009. What is the area of a circle whose diameter is exactly 12 meters?

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Your solution:

113m^2 (pi * 6m^2)

confidence rating #$&*: 3

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Given Solution:

`aThe area of a circle is A = pi r^2, where r is the radius. The radius of this circle is half the 12 m diameter, or 6 m. So the area is

A = pi ( 6 m )^2 = 36 pi m^2.

This result can be approximated to any desired accuracy by using a sufficient number of significant figures in our approximation of pi. For example using the 5-significant-figure approximation pi = 3.1416 we obtain A = 36 m^2 * 3.1416 = 113.09 m^2.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q010. What is the area of a circle whose circumference is 14 `pi meters?

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Your solution:

49 pi m^2 (C = 14 pi m= 2 pi r…..r = 7m….A=pi 7m^2)

confidence rating #$&*: 3

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Given Solution:

`aWe know that A = pi r^2. We can find the area if we know the radius r. We therefore attempt to use the given information to find r.

We know that circumference and radius are related by C = 2 pi r. Solving for r we obtain r = C / (2 pi). In this case we find that

r = 14 pi m / (2 pi) = (14/2) * (pi/pi) m = 7 * 1 m = 7 m.

We use this to find the area

A = pi * (7 m)^2 = 49 pi m^2.

STUDENT QUESTION:

Is the answer not 153.86 because you have multiply 49 and pi????

INSTRUCTOR RESPONSE

49 pi is exact and easier to connect to radius 7 (i.e., 49 is clearly the square of 7) than the number 153.86 (you can't look at that number and see any connection at all to 7).

You can't express the exact result with a decimal. If the radius is considered exact, then only 49 pi is an acceptable solution.

If the radius is considered to be approximate to some degree, then it's perfectly valid to express the result in decimal form, to an appropriate number of significant figures.

153.86 is a fairly accurate approximation.

However it's not as accurate as it might seem, since you used only 3 significant figures in your approximation of pi (you used 3.14). The first three figures in your answer are therefore significant (though you need to round); the .86 in your answer is pretty much meaningless.

If you round the result to 154 then the figures in your answer are significant and meaningful.

Note that a more accurate approximation (though still just an approximation) to 49 pi is 153.93804. An approximation to 5 significant figures is 153.94, not 153.86.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q011. What is the radius of circle whose area is 78 square meters?

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Your solution:

4.98m (78m^2 = pi r^2…..sqrt (78m^2/pi) = r)

confidence rating #$&*: 3

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Given Solution:

`aKnowing that A = pi r^2 we solve for r. We first divide both sides by pi to obtain A / pi = r^2. We then reverse the sides and take the square root of both sides, obtaining r = sqrt( A / pi ).

Note that strictly speaking the solution to r^2 = A / pi is r = +-sqrt( A / pi ), meaning + sqrt( A / pi) or - sqrt(A / pi). However knowing that r and A are both positive quantities, we can reject the negative solution.

Now we substitute A = 78 m^2 to obtain

r = sqrt( 78 m^2 / pi) = sqrt(78 / pi) m.{}

Approximating this quantity to 2 significant figures we obtain r = 5.0 m.

STUDENT QUESTION

Why after all the squaring and dividing is the final product just meters and not meters squared????

INSTRUCTOR RESPONSE

It's just the algebra of the units.

sqrt( 78 m^2 / pi) = sqrt(78) * sqrt(m^2) / sqrt(pi). The sqrt(78) / sqrt(pi) comes out about 5.

The sqrt(m^2) comes out m.

This is a good thing, since radius is measured in meters and not square meters.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q012. Summary Question 1: How do we visualize the area of a rectangle?

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Your solution:

We visualize the area of a rectangle by shading in the inside. We solve for the area of a rectangle by multiplying the length by the width. On a smaller scale, a rectangle can be broken into smaller one unit by one unit blocks with the number in length multiplied by the number of blocks in height.

confidence rating #$&*: 3

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Given Solution:

`aWe visualize the rectangle being covered by rows of 1-unit squares. We multiply the number of squares in a row by the number of rows. So the area is A = L * W.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q013. Summary Question 2: How do we visualize the area of a right triangle?

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Your solution:

A right triangle is ½ base * its height. Essentially it is visualized just like a rectangle and then divided by 2 since a right triangle would only be half.

confidence rating #$&*: 3

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Given Solution:

`aWe visualize two identical right triangles being joined along their common hypotenuse to form a rectangle whose length is equal to the base of the triangle and whose width is equal to the altitude of the triangle. The area of the rectangle is b * h, so the area of each triangle is 1/2 * b * h.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q014. Summary Question 3: How do we calculate the area of a parallelogram?

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Your solution:

We calculate the area of a parallelogram like we would a rectangle (as a parallelogram is essentially only a shifted rectangle), with its base times it’s altitude.

confidence rating #$&*: 3

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Given Solution:

`aThe area of a parallelogram is equal to the product of its base and its altitude. The altitude is measured perpendicular to the base.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q015. Summary Question 4: How do we calculate the area of a trapezoid?

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Your solution:

The area of a trapezoid is calculated by multiplying bases times the average of its altitude.

confidence rating #$&*: 3

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Given Solution:

`aWe think of the trapezoid being oriented so that its two parallel sides are vertical, and we multiply the average altitude by the width.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q016. Summary Question 5: How do we calculate the area of a circle?

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Your solution:

We calculate the area of a circle by multiplying pi times the radius^2.

confidence rating #$&*: 3

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Given Solution:

`aWe use the formula A = pi r^2, where r is the radius of the circle.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q017. Summary Question 6: How do we calculate the circumference of a circle? How can we easily avoid confusing this formula with that for the area of the circle?

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Your solution:

We calculate the circumference of a circle by multiplying 2 times pi times r. Area is squared units where circumference is not.

confidence rating #$&*: 3

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Given Solution:

`aWe use the formula C = 2 pi r. The formula for the area involves r^2, which will give us squared units of the radius. Circumference is not measured in squared units.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q018. Explain how you have organized your knowledge of the principles illustrated by the exercises in this assignment.

I have organized these principles in my head as these formulas are stored in my memory since they were drilled into me in high school.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

Self-critique (if necessary):

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Self-critique rating:

Self-critique (if necessary):

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#(*!

&#This looks good. Let me know if you have any questions. &#

course Phy 121

There are 12 questions in this document, along with some instructions. Copy this document into a word processor or text editor.

Answer the first two questions below, inserting your answers, Confidence Ratings and self-critiques as explained. You will then read the instructions that precede the remaining ten questions and answer those questions in a similar manner. Finally you will submit your work using a web form, according to instructions at the end of the document.

You will probably find that you can answer many of these questions without writing anything down. On those problems where you cannot arrive at an answer 'in your head', is recommended that you work out your solutions on paper. It is often helpful to sketch, doodle, jot down ideas, do calculations, organize and test ideas on paper.

When appropriate, you will later be encouraged to use a calculator to do any arithmetic you cannot do mentally. However the calculator is not appropriate to the questions that appear on this document. Put the calculator aside and think through these questions.

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Here is some additional information on the process and how it will fit into your course:

One of the predominant features of your course is the question-answer format for submitting work.

• In most courses you will encounter sequenced questions of this nature, designed to build your understanding by engaging you in the process of answering and self-critiquing your answers on a number of questions.

• In all courses you will submit assigned problems using this format.

As with the first couple of questions, the questions in this document can be answered with just a knowledge of basic high-school mathematics.

Sometimes the given solutions are more subtle than you might expect, and you will probably find that many of your answers, while good and correct, do not completely match the given solution. This is intentional, the goal being to get you used to the idea and the benefits of the self-critique process.

• Don't worry if you have trouble with a few of the questions, or if your explanations don't quite match those in the given solutions--most students begin their course a little rusty.

• Be sure to do your best to understand all the questions and the given solutions--it's this effort that makes the process beneficial to you.

The process is fairly simple, and you'll be using it again and again.

• The process will soon be very familiar to you, if it isn't already.

• Work through the instructions given here and within the questions, and do your best.

• If you miss something in the process (as most students are bound to do the first time through), your instructor will point it out to you, and there will be ample opportunity to get everything straight.

________________________________________

Your basic instructions follow. Rather than giving you the instructions at the beginning of the document, you were given a couple of questions to serve as a point of reference, and should now be better prepared to understand the instructions:

1. Answer each question, then look at the given solution:

It is expected that you will answer each question before looking at the given solution. There is no grade penalty for looking ahead, but if you do you:

• may be bypassing an opportunity to engage yourself in the solution process

• run the risk of deluding yourself about what you understand

• are likely to learn much less and

• are not as likely to do well in the course.

However your instructor understands the tradeoffs involved in being a student, and makes no judgement about how you should use this material. As long as you use it to your best advantage and succeed on tests and other course activities, you will get a good result from this course.

2. If you can't readily work it out in your head, use pencil and paper, and keep a record of your work.

You aren't expected to work out your solutions by staring at a computer screen, though you will likely find many questions and problems easy enough to do 'in your head'. However on more challenging problems, it's easier to work things out using a handwritten document than a computer-created or word-processed document.

• You should in general work out your answers to non-obvious problems on paper, jotting down sketches, diagrams and notes as you go, in such a way that you can make sense of it later. This will help you focus your work and maintain your train of thought, and will be quite valuable for periodic review. It is recommended (and may in some courses be required) that you dedicate a notebook to this course, and at least sketch out your work in the notebook.

3. There's no need for special formatting or graphs:

• Don't use special characters in your responses (e.g., characters like  that don't appear on your keyboard). The characters on your keyboard suffice to answer all these questions. If you use special characters they won't come through the form you use to submit your work, and if you use too many such characters your instructor might not be able to tell what you are saying.

• Don't try to make graphs in your document. Sketch your graphs by hand, then if necessary describe them in words (that probably won't be necessary in the present exercise; more about that later). Graphs won't come through when you submit your work. You can make a graph without understanding it, but you can't give a good description of your graph without understanding it. Your instructor doesn't need to see your graphs; he needs to see your descriptions of your graphs. The present exercise doesn't require extensive descriptions of graphs; it they apply to your course, you will see more about describing your graphs soon.

• When you submit this document (per instructions at the end), it will come to the instructor in pure text format. Any formatting you have done will not be seen by the instructor, special characters will not appear in what the instructor sees, and graphs won't come through. So don't do any fancy formatting for the instructor. You will of course want to save your original copy, and you are welcome to add formatting for your own purposes.

4. Expect to see your work posted by the instructor in a timely manner:

After submitting your work, per instructions at the end of this document, you should expect to see your work posted, along with instructor commentary, at your access page. It should be posted by the evening of the day after you submit it, and may well be posted the evening of the same day.

You have probably submitted your answers the first two questions in a preceding task. You are welcome to answer them again, but if you have already submitted them you may go ahead and skip to the third question.

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Question: `q001. If you are earning money at the rate of 8 dollars / hour and work for 4 hours, how much money do you make during this time? Answer in such a way as to explain your reasoning as fully as possible. A solution to this problem appears several lines below, but enter your own solution before you look at the given solution.

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Your solution: (type in your solution starting in the next line)

confidence rating #$&*:

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Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

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Given Solution: 8 dollars / hour means '8 dollars per hour', indicating that for every hour you work you earn 8 dollars. If you work for 4 hours, then if you earn 8 dollars for every one of those hours you earn 4 * 8 dollars = 32 dollars.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

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Self-critique Rating:

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

*********************************************

Question: `q002. If you work 12 hours and earn $168, then at what rate, in dollars / hour, were you making money?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: $168 earned in 12 hours implies that $168 / 12 = $14 were made per hour, so the rate is $14 / hour.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

------------------------------------------------

Self-critique Rating:

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

________________________________________

Here are the remaining ten questions:

*********************************************

Question: `q003. If you are earning 8 dollars / hour, how long will it take you to earn $72? The answer may well be obvious, but explain as best you can how you reasoned out your result.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

You receive $8 each hour you work, so for 2 hours you receive $16, 3 hours $24, and so on. Dividing $72 by $8 gives you 9 hours work needed to receive $72.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

.............................................

Given Solution: Many students simply know, at the level of common sense, that if we divide $72 by $8 / hour we get 9 hours, so 9 hours are required.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

------------------------------------------------

Self-critique Rating: OK

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

*********************************************

Question: `q004. Calculate (8 + 3) * 5 and 8 + 3 * 5, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

(8+3)*5 = 11*5 = 55 Do the operation in the parentheses first and then multiply.

8+3*5 = 8+15 = 23 Do the multiplication first and then add.

""Please excuse my dear Aunt Sally"" indicates the order of operations.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: (8 + 3) * 5 and 8 + 3 * 5

To evaluate (8 + 3) * 5, you will first do the calculation in parentheses. 8 + 3 = 11, so

(8 + 3) * 5 = 11 * 5 = 55.

To evaluate 8 + 3 * 5 you have to decide which operation to do first, 8 + 3 or 3 * 5. You should be familiar with the order of operations, which tells you that multiplication precedes addition. The first calculation to do is therefore 3 * 5, which is equal to 15. Thus

8 + 3 * 5 = 8 + 15 = 23

The results are different because the grouping in the first expression dictates that the addition be done first.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

------------------------------------------------

Self-critique Rating: OK

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

In subsequent problems the detailed instructions that accompanied the first four problems are missing. We assume you will know to follow the same instructions in answering the remaining questions.

*********************************************

Question: `q005. Calculate (2^4) * 3 and 2^(4 * 3), indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. Note that the symbol '^' indicates raising to a power. For example, 4^3 means 4 raised to the third power, which is the same as 4 * 4 * 4 = 64.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

(2^4)*3 = 16*3 = 48 Raise 2 to the 4th power as indicated inside the parenthesis and the multiply by 3.

2^(4*3) = 2^12 = 4096 Do the multiplication inside the parenthesis and then raise 2 to the 12th power.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution:

To evaluate (2^4) * 3 we first evaluate the grouped expression 2^4, which is the fourth power of 2, equal to 2 * 2 * 2 * 2 = 16. So we have

(2^4) * 3 = 16 * 3 = 48.

To evaluate 2^(4 * 3) we first do the operation inside the parentheses, obtaining 4 * 3 = 12. We therefore get

2^(4 * 3) = 2^12 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 4096.

It is easy to multiply by 2, and the powers of 2 are important, so it's appropriate to have asked you to do this problem without using a calculator. Had the exponent been much higher, or had the calculation been, say, 3^12, the calculation would have become tedious and error-prone, and the calculator would have been recommended.

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Self-critique (if necessary):OK

*********************************************

Question: `q006. Calculate 3 * 5 - 4 * 3 ^ 2 and 3 * 5 - (4 * 3)^2 according to the standard order of operations, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

3*5-4*3^2 = 3*5-4*9 = 15-36 = -21 First, raise 3 to the 2nd power, do the multiplication and then subtract.

3*5-(4*3)^2 = 3*5-12^2 = 3*5-144 = 15-144 = -129 Clear parentheses, raise 12 to the 2nd power, multiply and then subtract.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

To calculate 3 * 5 - 4 * 3 ^ 2, the first operation is the exponentiation operation ^.

• The two numbers involved in the exponentiation are 3 and 2; the 4 is 'attached' to the 3 by multiplication, and this multiplication can't be done until the exponentiation has been performed.

• The exponentiation operation is therefore 3^2 = 9, and the expression becomes 3 * 5 - 4 * 9.

Evaluating this expression, the multiplications 3 * 5 and 4 * 9 must be performed before the subtraction. 3 * 5 = 15 and 4 * 9 = 36 so we now have

3 * 5 - 4 * 3 ^ 2 = 3 * 5 - 4 * 9 = 15 - 36 = -21.

To calculate 3 * 5 - (4 * 3)^2 we first do the operation in parentheses, obtaining 4 * 3 = 12. Then we apply the exponentiation to get 12 ^2 = 144. Finally we multiply 3 * 5 to get 15. Putting this all together we get

3 * 5 - (4 * 3)^2 =

3 * 5 - 12^2 =

3 * 5 - 144 =

15 - 144 =

-129.

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Self-critique (if necessary):OK

------------------------------------------------

Self-critique Rating:3

In the next three problems, the graphs will be of one of the basic shapes listed below. You will be asked to construct graphs for three simple functions, and determine which of the depicted graphs each of your graphs most closely resembles. At this point you won't be expected to know these terms or these graph shapes; if at some point in your course you are expected to know these things, they will be presented at that point.

Linear:

Quadratic or parabolic:

Exponential:

Odd power:

Fractional positive power:

Even negative power:

partial graph of polynomial of degree 3

more extensive graph of polynomial of degree 3

*********************************************

Question: `q007. Let y = 2 x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

-2 -1

-1 1

0 3

1 5

2 7

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

To obtain the y values, substitute the x value in the equation, y=2x+3, and solve. The y values increase the same when x is increased the same so the graph is linear. All the pairs when plotted will be a straight line.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Two slightly different explanations are give below, one by a student and one by the instructor. Neither format is inherently better than the other.

GOOD SOLUTION BY STUDENT:

First we need to complete the table. I have added a column to the right of the table to show the calculation of “y” when we us the “x” values as given.

x y Calculation: If y = 2x + 3

-2 -1 If x = -2, then y = 2(-2)+3 = -4+3 = -1

-1 1 If x= -1, then y = 2(-1)+3 = -2+3 = 1

0 3 If x= 0, then y = 2(0)+3 = 0+3 = 3

1 5 If x= 1, then y = 2(1)+3 = 2+3 = 5

2 7 If x= 2, then y = 2(2)+3 = 4+3 = 7

Once an answer has been determined, the “y” value can be filled in. Now we have both the “x” and “y” values and we can begin our graph. The charted values continue on a straight line representing a linear function as shown above.

INSTRUCTOR'S SOLUTION:

We easily evaluate the expression:

• When x = -2, we get y = 2 x + 3 = 2 * (-2) + 3 = -4 + 3 = -1.

• When x = -1, we get y = 2 x + 3 = 2 * (-1) + 3 = -2 + 3 = 1.

• When x = 0, we get y = 2 x + 3 = 2 * (0) + 3 = 0 + 3 = 3.

• When x = 1, we get y = 2 x + 3 = 2 * (1) + 3 = 2 + 3 = 5.

• When x = 2, we get y = 2 x + 3 = 2 * (2) + 3 = 4 + 3 = 7.

Filling in the table we have

x y

-2 -1

-1 1

0 3

1 5

2 7

When we graph these points we find that they lie along a straight line.

Only one of the depicted graphs consists of a straight line, and we conclude that the appropriate graph is the one labeled 'linear'.

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Self-critique (if necessary):OK

------------------------------------------------

Self-critique Rating:3

*********************************************

Question: `q008. Let y = x^2 + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

-2 7

-1 4

0 3

1 4

2 7

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The y values are calculated by substituting the x value in the equation, y=x^2+3, and solving. The two pairs are the same except the x value in the first pair is negative and positive in the second indicating the graph crosses the y axis and forms a parabola.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Evaluating y = x^2 + 3 at the five points:

• If x = -2 then we obtain y = x^2 + 3 = (-2)^2 + 3 = 4 + 3 = 7.

• If x = -1 then we obtain y = x^2 + 3 = (-1)^2 + 3 = ` + 3 = 4.

• If x = 0 then we obtain y = x^2 + 3 = (0)^2 + 3 = 0 + 3 = 3.

• If x = 1 then we obtain y = x^2 + 3 = (1)^2 + 3 = 1 + 3 = 4.

• If x = 2 then we obtain y = x^2 + 3 = (2)^2 + 3 = 4 + 3 = 7.

The table becomes

x y

-2 7

-1 4

0 3

1 4

2 7

We note that there is a symmetry to the y values. The lowest y value is 3, and whether we move up or down the y column from the value 3, we find the same numbers (i.e., if we move 1 space up from the value 3 the y value is 4, and if we move one space down we again encounter 4; if we move two spaces in either direction from the value 3, we find the value 7).

A graph of y vs. x has its lowest point at (0, 3).

If we move from this point, 1 unit to the right our graph rises 1 unit, to (1, 4), and if we move 1 unit to the left of our 'low point' the graph rises 1 unit, to (-1, 4).

If we move 2 units to the right or the left from our 'low point', the graph rises 4 units, to (2, 7) on the right, and to (-2, 7) on the left.

Thus as we move from our 'low point' the graph rises up, becoming increasingly steep, and the behavior is the same whether we move to the left or right of our 'low point'. This reflects the symmetry we observed in the table. So our graph will have a right-left symmetry.

Two of the depicted graphs curve upward away from the 'low point'. One is the graph labeled 'quadratic or parabolic'. The other is the graph labeled 'partial graph of degree 3 polynomial'.

If we look closely at these graphs, we find that only the first has the right-left symmetry, so the appropriate graph is the 'quadratic or parabolic' graph.

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Self-critique (if necessary):OK

------------------------------------------------

Self-critique Rating:3

*********************************************

Question: `q009. Let y = 2 ^ x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = 1. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values 2, 3 and 4. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

1 5

2 7

3 11

4 19

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Substituting the x value in the equation gives us the y values. The x values increase at the same rate and the y values increase at increasingly greater rates indicating the graph will increase as it goes from left to right. It is therefore exponential.

Order of operations not indicated####

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Recall that the exponentiation in the expression 2^x + 1 must be done before, not after the addition.

When x = 1 we obtain y = 2^1 + 3 = 2 + 3 = 5.

When x = 2 we obtain y = 2^2 + 3 = 4 + 3 = 7.

When x = 3 we obtain y = 2^3 + 3 = 8 + 3 = 11.

When x = 4 we obtain y = 2^4 + 3 = 16 + 3 = 19.

x y

1 5

2 7

3 11

4 19

Looking at the numbers in the y column we see that they increase as we go down the column, and that the increases get progressively larger. In fact if we look carefully we see that each increase is double the one before it, with increases of 2, then 4, then 8.

When we graph these points we find that the graph rises as we go from left to right, and that it rises faster and faster. From our observations on the table we know that the graph in fact that the rise of the graph doubles with each step we take to the right.

The only graph that increases from left to right, getting steeper and steeper with each step, is the graph labeled 'exponential'.

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Self-critique (if necessary):OK

------------------------------------------------

Self-critique Rating:3

*********************************************

Question: `q010. If you divide a certain positive number by 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The result is equal to the original.5/1=5, 25/1=25, 50/1=50

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution: If you divide any number by 1, the result is the same as the original number. Doesn't matter what the original number is, if you divide it by 1, you don't change it.

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Self-critique (if necessary):OK

------------------------------------------------

Self-critique Rating:3

*********************************************

Question: `q011. If you divide a certain positive number by a number greater than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Less than the original number. 30/1=30, 30/2=15, 30/ 6=5

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by another number is similar. The bigger the number you divide by, the less you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a number greater than 1, what you get has to be smaller than the original number. Again it doesn't matter what the original number is, as long as it's positive.

Students will often reason from examples. For instance, the following reasoning might be offered:

OK, let's say the original number is 36. Let's divide 36 be a few numbers and see what happens:

36/2 = 18. Now 3 is bigger than 2, and

36 / 3 = 12. The quotient got smaller. Now 4 is bigger than 3, and

36 / 4 = 9. The quotient got smaller again. Let's skip 5 because it doesn't divide evenly into 36.

36 / 6 = 4. Again we divided by a larger number and the quotient was smaller.

I'm convinced.

That is a pretty convincing argument, mainly because it is so consistent with our previous experience. In that sense it's a good argument. It's also useful, giving us a concrete example of how dividing by bigger and bigger numbers gives us smaller and smaller results.

However specific examples, however convincing and however useful, don't actually prove anything. The argument given at the beginning of this solution is general, and applies to all positive numbers, not just the specific positive number chosen here.

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Self-critique (if necessary):OK

------------------------------------------------

Self-critique Rating:3

*********************************************

Question: `q012. If you divide a certain positive number by a positive number less than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Greater. 10/0.5=20, 10/0.1=100, 10/0.2=50

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by some other number is similar. The bigger the number you divide by, the less you get. The smaller the number you divide by, the more you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a positive number less than 1, what you get has to be larger than the original number. Again it doesn't matter what the original number is, as long as it's positive.

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Self-critique (if necessary):OK

------------------------------------------------

Self-critique Rating:3

*********************************************

Question: `q013. Students often get the basic answers to nearly all, or even all these questions, correct. Your instructor has however never seen anyone who addressed all the subtleties in the given solutions in their self-critiques, and it is very common for a student to have given no self-critiques. It is very likely that there is something in the given solutions that is not expressed in your solution.

This doesn't mean that you did a bad job. If you got most of the 'answers' right, you did fine.

However, in order to better understand the process, you are asked here to go back and find something in one of the given solutions that you did not address in your solution, and insert a self-critique. You should choose something that isn't trivial to you--something you're not 100% sure you understand.

If you can't find anything, you can indicate this below, and the instructor will point out something and request a response (the instructor will select something reasonable, but will then expect a very good and complete response). However it will probably be less work for you if you find something yourself.

Your response should be inserted at the appropriate place in this document, and should be indicated by preceding it with ####.

As an answer to this question, include a copy of whatever you inserted above, or an indication that you can't find anything.

your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

Order of operations not indicated.

________________________________________