course Mth 163 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Same Self-critique Rating: 3 ********************************************* Question: `q002. What is the area of a right triangle whose legs are 4.0 meters and 3.0 meters? ********************************************* Your solution: Area of a right triangle is half that of the rectangle that shares the legs as two sides of the rectangle. Use the hypotenuse of each triangle as a line bisecting the rectangle with the right angles opposite one another. Area of the outer rectangle = 4 m * 3 m = 12 m ^2 Area of each of the individual right triangles = 12 m^2 / 2 = 6m ^2 Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I did not remember the formula for the right triangle, but it works logically from my assumption about it being half the area of the rectangle. Self-critique Rating: 3 ********************************************* Question: `q003. What is the area of a parallelogram whose base is 5.0 meters and whose altitude is 2.0 meters? ********************************************* Your solution: The area of a parallelogram can be calculated from the area of a rectangle that has legs the same size. If you drop a line from each point of the parallelogram inside the figure down to the opposing side at a right angle on the opposing side, it defines a right triangle. The area of the right triangle inside the shape is exactly the same as the area of the right triangle “missing” on the outside of the parallelogram to turn the parallelogram into a rectangle. (move it from one side to the opposing side, virtually, and you an color in a rectangle) The base is 5, therefore the inner rectangle is (5 less the altitude)m * (altitude)m = (5-2)m * 2m = 3m * 2m =6m^2 The rectangle formed by the triangles on either end, are (altitude)m * (altitude)m = = 2 m * 2 m = 4 m^2 Base rectangle + outer rectangle = 6m^2 + 4m^2 Area of parallelogram = 10m^2 Confidence Assessment: 3
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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): Once again I did not remember the formula, but the answer works out as described. Self-critique Rating: 3 ********************************************* Question: `q004. What is the area of a triangle whose base is 5.0 cm and whose altitude is 2.0 cm? ********************************************* Your solution: The area of the triangle is half the area of the rectangle defined by legs the same size. The triangle will fit into a rectangle with a matching altitude and base. Divide the rectangle with line from the point to the base, at a right angle. The area of each of these inner triangles is now equal, and each compliments the other by aligning the hypotenuse to form a rectangle exactly half the area of the original outer rectangle. Outer rectangle = 5 cm * 2 cm = 10 cm^2 Inner rectangle = 10cm^2 / 2 = 5cm^2 Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Different technique was used. I see how technique of combining two triangles would also work Self-critique Rating: ********************************************* Question: `q005. What is the area of a trapezoid with a width of 4.0 km and average altitude of 5.0 km? ********************************************* Your solution: A trapezoid has two parallel sides, of four total sides. Although the total of all four angles is 360⁰, no right angles can be assumed. One form can be defined by an internal rectangle (which may or may not be square) and a right angle rectangle on one end. The other (which contains two right angles in the original trapezoid) can be defined as an internal rectangle (which still may or may not be square) and one right angle rectangle. In either case, the area of the trapezoid will equal the area of the internal rectangle, plus half the area of the external rectangle created by squaring off the outside triangle or triangles. If the short parallel side (lesser of the two altitude measurements) is represented as “a” and the longer (greater of the two altitude measurements) is represented as “b” , and “c” is the difference between the two then the area is found by = (a * w) + [1/2(b-a) * w] Now because b = a+c , substitute = [a + (1/2((a+c)-a)]* w =[a + (1/2(c )] * w = (a +.5*c) * w Now (a+ .5*c) = [(a+ .5*c) + (a+ .5*c)]/2 = [a + a + .5*c + .5c] / 2 = [ a + a + c ] /2 = [ a + (a+c)] / 2 = (a+b)/2 also know as average altitude Therefore the area of the trapezoid = average altitude * width = 5 km * 4 km = 20 km^2 Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Same Self-critique Rating: 3 ********************************************* 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? ********************************************* Your solution: Given the same reasoning I presented on the question q005, trapezoid area = average altitude * width = (3cm + 8cm)/ 2 * 4cm =11cm / 2 * 4 cm = 5.5 cm * 4 cm = 22 cm^2 Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Same Self-critique Rating:3 ********************************************* Question: `q007. What is the area of a circle whose radius is 3.00 cm? ********************************************* Your solution: Area of a circle formula is pi*r^2 (followed by the old joke,”No it ain’t! Pie are round! - sorry) With two decimals in the radius measurement, I used only two in the approximation for pi, therefore Pi * r^ 2 ≈ 3.14 * 3.00 cm^2 ≈ 3.14 * 9 cm^2 ≈ 28.26 cm^2 Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OOPS – I had confused the concept of degree of accuracy with significant figures. I reviewed the concept (http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/sigfigs5.html) and of course remembered “In multiplication and division, the result should be rounded off so as to have the same number of significant figures as in the component with the least number of significant figures”. In this case, 3.00 has three significant figures (zeros to the right of the decimal are counted) so the answer should have been rounded to three significant figures as well, in this case 28.26 cm^2 is more appropriately rounded to 28.3 cm^2 Self-critique Rating: 3 ********************************************* Question: `q008. What is the circumference of a circle whose radius is exactly 3 cm? ********************************************* Your solution: The circumference = 2* pi * radius = 2 * pi * 3cm = 6* pi cm (exact value) Approximating pi ≈ 6 * 3.14 cm ≈ 18.84 cm Using three significant digits ≈ 18.8 cm Confidence Assessment: 3
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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 = 18.84 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): QUESTION********* OK, I am having a problem with the appropriate number of significant figures in the answer, How can we use 3.14 as the value of pie in the calculation, and have four significant figures in the answer? Self-critique Rating: 2 ********************************************* Question: `q009. What is the area of a circle whose diameter is exactly 12 meters? ********************************************* Your solution: Diameter = 2 * radius 12 m = 2 * radius 12m / 2 = radius 6m = radius Area = pi*r^2 = pi * 6m ^ 2 = pi * 36m^2 (exact) ≈ 3.14 * 36m^2 (approximation) ≈ 113.04 m^2 ≈ 113 m^2 (to three significant figures) Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK Self-critique Rating: 3 ********************************************* Question: `q010. What is the area of a circle whose circumference is 14 `pi meters? ********************************************* Your solution: C = 2*pi * r 14* pi meters = 2 * pi *r Divide both sides by 2*pi 7 meters = r A = pi * r ^2 = pi * 7m^2 = pi * 49m^2 ≈ 3.14 * 49m^2 ≈ 153.86 m^2 ≈154 m^2 Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Matches last exact figure I gave, I further provided approximation based on pi = 3.14 Self-critique Rating: 3 ********************************************* Question: `q011. What is the radius of circle whose area is 78 square meters? ********************************************* Your solution: A = pi * r^2 78 m^2 = pi * r^2 (78/pi)m^2 = r^2 √(78/pi)m = r √(78/3.14)m ≈ r √(24.84) m ≈ r √(24.84) m ≈ r (radius cannot be negative length, therefore positive value only) 4.98 m ≈ r 5.0 m ≈ r (two significant figures, like the original 78) Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: 3 ********************************************* Question: `q012. Summary Question 1: How do we visualize the area of a rectangle? ********************************************* Your solution: Grid the rectangle into units, length by width (cm, inches, whatever as long as they are the same both ways). The area is the number of resulting boxes. Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: 3 ********************************************* Question: `q013. Summary Question 2: How do we visualize the area of a right triangle? ********************************************* Your solution: Complete the right triangle into a rectangle by extending a line from each of the points (not the right angle) perpendicular to the side that ends at each point. The resulting rectangle is exactly twice the size of the original triangle. Use the same grid technique (length by width of the rectangle) and count the boxes (or multiply length by width) then divide in half for the area of each rectangle. Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Same Self-critique Rating: 3 ********************************************* Question: `q014. Summary Question 3: How do we calculate the area of a parallelogram? ********************************************* Your solution: The area of a parallelogram is the same as the area of a rectangle with one side the length of one of the parallel sides, and the other side the altitude of the parallelogram. Draw a line from each end of one side of a parallelogram, ending at the extension of the opposite side of the parallelogram. This is the altitude. The “missing” triangle on one side of the resulting rectangle is exactly equal to the “extra” triangle on the other side. Therefore altitude times base will give the number of units of area in the parallelogram, just as for the rectangle. Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): same (but mine has pictures) ;-) Self-critique Rating: 3 ********************************************* Question: `q015. Summary Question 4: How do we calculate the area of a trapezoid? ********************************************* Your solution: Average altitude times width (a1 + a2)/2 * w Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): same Self-critique Rating:3 ********************************************* Question: `q016. Summary Question 5: How do we calculate the area of a circle? ********************************************* Your solution: A = pi*r^2 Confidence Assessment: 3
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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): same Self-critique Rating: ********************************************* 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? ********************************************* Your solution: C = 2*pi*r 1) Circumference is measured in units, Area is measured in square units, If the answer is in the wrong units, the wrong formula was used. Confidence Assessment: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): same Self-critique Rating: 3 ********************************************* Question: `q018. Explain how you have organized your knowledge of the principles illustrated by the exercises in this assignment. This exercise is a review of area/volume calculations for different geometric shapes. The exercise has been retained with drawings inserted as a word file, and I have dictated .wav files as a refresher to play during the day to “reset” the formulas in my memory. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Only unsettled element is the use of significant figures – specifically on the solution given for q008
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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 = 18.84 cm. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): QUESTION********* OK, I am having a problem with the appropriate number of significant figures in the answer, How can we use 3.14 as the value of pie in the calculation, and have four significant figures in the answer? Self-critique Rating: 2