#$&* course Mth 158 i am not completely sure why the first attempt was not appropriate. to ensure that it will work properly i have done as you asked and copied the document from the homepage in its entirety and pasted my answers where appropriate. documentshort description of contentwhat you'll know when you're doneAreasreviews the meaning, reasoning and calculation of areas of some common geometric figuresthe meaning of areas, reasoning about areas, some important formulas for areas
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q002.What is the area of a right triangle whose legs are 4.0 meters and 3.0 meters? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find the area of a triangle we will use the formula to do so. The formula is base*height /2=area. So to do this we will multiply 3 by 4 and divide by two. 3*4=12 then 12/2=6. So the answer is a triangle with legs with the length of 3 and 4 will have an area of 6 meters squared. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q003.What is the area of a parallelogram whose base is 5.0 meters and whose altitude is 2.0 meters? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To obtain the area for a parallelogram we take base and altitude and multiply them. This will give the area (2*5=10) so the area would be 10 meters squared. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aA parallelogram is easily rearranged into a rectangle by 'cutting off' the protruding end, turning that portion upside down and joining it to the other end.Hopefully you are familiar with this construction.In any case the resulting rectangle has sides equal to the base and the altitude so its area is A = b * h. The present rectangle has area A = 5.0 m * 2.0 m = 10 m^2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I honestly was just doing the same formula that is done for a rectangle, I couldn’t remember if there was a specific formula for parallelograms, so I guessed. But now that I look at the solution it makes sense why it’s done this way. Just like the triangle if you think about a shape in relation to a rectangle then it becomes much more clear and easy. Self-critiqueRating:ok ********************************************* Question:`q004.What is the area of a triangle whose base is 5.0 cm and whose altitude is 2.0 cm? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find the area of a triangle we multiply the base by the altitude and then divide by 2. In this case you would multiply 2 by 5 (2*5=10) then divide by two 10/2=5. So the answer would be area=5 cm^2 confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q005.What is the area of a trapezoid with a width of 4.0 km and average altitude of 5.0 km? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To solve this problem we take the base and multiply it by height. So the equation would be 4*5=20. So the area of this trapezoid would be 20 km^2. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aAny trapezoid can be reconstructed to form a rectangle whose width is equal to that of the trapezoid and whose altitude is equal to the average of the two altitudes of the trapezoid.The area of the rectangle, and therefore the trapezoid, is therefore A = base * average altitude.In the present case this area is A = 4.0 km * 5.0 km = 20 km^2. STUDENT SOLUTION ILLUSTRATING NEED TO USE UNITS IN ALL STEPS A=Base time average altitude therefore………A=4 *5= 20 km ^2 INSTRUCTOR COMMENT A = (4 km) * (5 km) = 20 km^2. Use the units at every step. km * km = km^2, and this is why the answer comes out in km^2. Try to show the units and how they work out in every step of the solution. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): In previous problems, I have not used the unit that the problem was given in in all steps of my answer. I will do so from now on. Self-critiqueRating:ok ********************************************* Question:`q006.What is the area of a trapezoid whose width is 4 cm in whose altitudes are 3.0 cm and 8.0 cm? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given that the area of a trapezoid is found the same way a rectangle is, and given the fact that there are two height measurements we need to find the average height before we continue with finding the area. So to find the average height we take both height values and add them then divide by the number of values. 3.0cm+8.0cm=11.0cm then 11.0cm/2=5.5cm. so the average height is 5.5cm. now we take the base and multiply it by the height (4cm*5.5cm=22cm^2) so the area of this trapezoid is 22cm^2. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q007.What is the area of a circle whose radius is 3.00 cm? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find the area of a circle we use the equation a=2 pi r, so the area of this circle is 6 pi cm^2. (a=2 pi 3=6) confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe area of a circle is A = pi * r^2, where r is the radius.Thus A = pi * (3 cm)^2 = 9 pi cm^2. Note that the units are cm^2, since the cm unit is part r, which is squared. The expression 9 pi cm^2 is exact.Any decimal equivalent is an approximation.Using the 3-significant-figure approximation pi = 3.14 we find that the approximate area is A = 9 pi cm^2 = 9 * 3.14 cm^2 = 28.26 cm^2, which we round to 28.3 cm^2 to match the number of significant figures in the given radius. Be careful not to confuse the formula A = pi r^2, which gives area in square units, with the formula C = 2 pi r for the circumference.The latter gives a result which is in units of radius, rather than square units.Area is measured in square units; if you get an answer which is not in square units this tips you off to the fact that you've made an error somewhere. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I did exactly what you warned in the given solution. I was trying to use the formula to find the circumference. But now I know the difference. C=2 pi r where as a=pi*r^2. Self-critiqueRating:ok ********************************************* Question:`q008.What is the circumference of a circle whose radius is exactly 3 cm? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given that the formula for circumference is c=2*pi*r, the circumference of this circle will be 6pi cm. because the radius (3cm) multiplied by 2 gives us six and also multiplied times pi, which as before stated is exact and any decimal value will be skewed. Therefore the circumference is 6pi cm. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q009.What is the area of a circle whose diameter is exactly 12 meters? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find the area of this circle we need to find the radius. To do that we simply divide the diameter by 2 (12m/2=6m), now that we have the radius we place the values in the given equation for area. (a=pi*r^2) so 6m^2 is 36m this gives us the answer 36pi m^2. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe area of a circle is A = pi r^2, where r is the radius.The radius of this circle is half the 12 m diameter, or 6 m.So the area is A = pi( 6m )^2 = 36 pi m^2. This result can be approximated to any desired accuracy by using a sufficient number of significant figures in our approximation of pi.For example using the 5-significant-figure approximation pi = 3.1416 we obtain A = 36 m^2 * 3.1416 = 113.09 m^2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q010.What is the area of a circle whose circumference is 14 `pi meters? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To perform this problem we must work the formula for the circumference backwards to discover what the radius is. After we figure out what the radius is, (its 7 because c=2*pi*7=14 pi m), we then can calculate the area. With a radius of 7 we will calculate a=pi*r^2 giving us 49 pi m^2. So the area of this circle is 49 pi m^2. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q011.What is the radius of circle whose area is 78 square meters? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Since the formula for area is a=p*r^2 we will work this backwards to solve for the radius. We assume that since the denotation of pi is not present in the problem above, the radius squared has been multiplied by the three place decimal equivalent of pi. (3.14) therefore we divide 78m^2 by 3.14 which gives us 24.84. Then we take the square root of this result which gives us 4.98 which when rounded becomes 5.0 m. therefore the radius of this circle is 5 meters. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aKnowing that A = pi r^2 we solve for r.We first divide both sides by pi to obtain A / pi = r^2.We then reverse the sides and take the square root of both sides, obtaining r =sqrt(A / pi ). Note that strictly speaking the solution to r^2 = A / pi is r = +-sqrt(A / pi ), meaning + sqrt( A / pi) or - sqrt(A / pi).However knowing that r andAare both positive quantities, we can reject the negative solution. Now we substitute A = 78 m^2 to obtain r =sqrt(78 m^2 / pi) = sqrt(78 / pi) m.{} Approximating this quantity to 2 significant figures we obtain r = 5.0 m. STUDENT QUESTION Why after all the squaring and dividing is the final product just meters and not meters squared???? INSTRUCTOR RESPONSE It's just the algebra of the units. sqrt( 78 m^2 / pi) = sqrt(78) * sqrt(m^2) / sqrt(pi). The sqrt(78) / sqrt(pi) comes out about 5. The sqrt(m^2) comes out m. This is a good thing, since radius is measured in meters and not square meters. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q012.Summary Question 1:How do we visualize the area of a rectangle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We visualize the area of a rectangle as the space inside the perimeter of the shape calculated in square units. This is done by taking the base and multiplying it by the height. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aWevisualizethe 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q013.Summary Question 2:How do we visualize the area of a right triangle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We visualize the area of a right triangle by understanding that of a rectangle. When two right triangles of equal size and shape are organized in such a way they make a rectangle. Thus we know to find the area of a rectangle we multiply base by height, however this will give us the area of two of the triangle, therefore we divide our total area by 2 giving us the area of our original triangle. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aWevisualizetwo identical right triangles being joined along their common hypotenuse to form a rectangle whose length is equal to the base of the triangle and whose width is equal to the altitude of the triangle.The area of the rectangle is b * h, so the area of each triangle is 1/2 * b * h. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q014.Summary Question 3:How do we calculate the area of a parallelogram? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To calculate the area of a parallelogram, all we need to do is rearrange the shape into the confines of a rectangle, after we have done that we simply solve it like it was a rectangle. A=b*h confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q015.Summary Question 4:How do we calculate the area of a trapezoid? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To calculate the area of a trapezoid we need to take the base and multiply by the height. If there is more than one value for the height we take an average of these two heights and then solve for area. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aWethinkof the trapezoid being oriented so that its two parallel sides are vertical, and we multiply the average altitude by the width. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q016.Summary Question 5:How do we calculate the area of a circle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To calculate the area of a circle we simply follow the formula a=pi*r^2. We multiply the radius by the radius and then by pi. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aWe use the formula A = pi r^2, where r is the radius of the circle. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find the circumference of a circle we follow the given formula c=2*pi*r. this will give us the answer for the circumference of a circle. The easiest way to distinguish between these two formulas is the formula for the area of a circle has the radius squared and the answer will be in square units where as the circumference will not. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok Self-critiqueRating:ok ********************************************* Question:`q018.Explain how you have organized your knowledge of the principles illustrated by the exercises in this assignment. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I have organized this info/knowledge in this way: the area of a rectangle is the building block for calculating the area of a trapezoid, parallelogram, and right triangles. They are closely related and be rearranging the shape to make a rectangle they can be easily solved. Also the area and perimeter and area of a circle both have formulas to follow and the most important part is distinguishing between the two. Self-critiqueRating:ok " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: