course Mth163 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
.............................................
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): Remember that in my problem A =4m * 3m That I am multiplying (4 * 3) (m * m) Thus giving me 12 * m^2 or 12m^2 Self-critique Rating: ********************************************* Question: `q002. What is the area of a right triangle whose legs are 4.0 meters and 3.0 meters? ********************************************* Your solution: The formula for the area of this triangle is A = ½ *b * h , because of the right angles we must take ½ of the equation to find the area. of this half of triangle. So A = ½ * 4 m * 3 m = A = ½ * 12m^2 = A = 6 m ^ 2 Confidence Assessment: positive
.............................................
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): ok Self-critique Rating: ********************************************* Question: `q003. What is the area of a parallelogram whose base is 5.0 meters and whose altitude is 2.0 meters? ********************************************* Your solution: To find the area of this parallelogram you need to solve this formula: A = b * h = A = 5 m * 2 m = A = 10 m ^2 Confidence Assessment: positive
.............................................
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): ok Self-critique Rating: ********************************************* Question: `q004. What is the area of a triangle whose base is 5.0 cm and whose altitude is 2.0 cm? ********************************************* Your solution: A = ½ * b * h = A = ½ * 5.cm * 2 cm = A = ½ * 10 cm^2 = A = 5 cm^2 Confidence Assessment:
.............................................
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 appropriate 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-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: To find the area for this trapezoid you would have to do this formula: A = W * altitude = A = 4 m * 5 m = A = 20 m^2 Confidence Assessment: mostly
.............................................
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): ok Self-critique Rating: ********************************************* 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: In order to find the area of this trapezoid you would have to find the average of the altitudes, (3cm + 8cm) / 2 = 11cm / 2 = 5.5 cm Them do A = w * altitude = A = 4cm * 5.5cm = 22 cm^2 Confidence Assessment: mostly
.............................................
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-critique Rating: ********************************************* Question: `q007. What is the area of a circle whose radius is 3.00 cm? ********************************************* Your solution: A = pi * r^2 since the radius is 3.00 cm then we substitute that into the problem A = pi * 3cm^2 = pi * 9cm^2 since pi is estimated at 3.14 then we substitute that into the equation A = 3.14 * 9cm^2 = 28.26cm^2 Confidence Assessment: mostly
.............................................
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): ok Self-critique Rating: ********************************************* Question: `q008. What is the circumference of a circle whose radius is exactly 3 cm? ********************************************* Your solution: the formula for the circumference of a circle is: c = 2 pi r C = 2 pi * 3cm The given radius was 3cm, I just substituted it in the equation C = 6 pi cm C = 6 * 3.14 cm The estimated number for pi is 3.14 C = 18.84cm Confidence Assessment: mostly
.............................................
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): Remember if question ask for approximation or if statement implies exact numbers. Self-critique Rating: ********************************************* Question: `q009. What is the area of a circle whose diameter is exactly 12 meters? ********************************************* Your solution: A = pi * r^2 If the radius is ½ of the diameter then the radius for this circle is 6m A = pi * (6m)^2 A = 36pi m^2 Confidence Assessment: positive
.............................................
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): I didn’t do pi because of the approximation and my problem gave the exact amount Self-critique Rating: ********************************************* Question: `q010. What is the area of a circle whose circumference is 14 `pi meters? ********************************************* Your solution: I know that to find area of a circle you must do A = pi*r^2 but without a radius this can’t be used I know that the circumference is found using C = 2pi r , without the radius I’m not sure how to figure out how to find it Confidence Assessment: unsure
.............................................
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): I needed to find the radius using the circumference formula C = 2pi r move 2pi to other side in order to solve for r r = c / (2pi) we know that c =14 pi m r = 14 pi m / (2pi) you could group the division better to understand better r = (14/2 )*( pi / pi )m = 7 * 1m = m now we can find the area since we have the radius A = pi * r^2 A = pi * (7m)^2 = 49 pi m^2 Self-critique Rating: ********************************************* Question: `q011. What is the radius of circle whose area is 78 square meters? ********************************************* Your solution: A = pi * r^2 78m^2 = pi *r^2 divide both sides by pi 78m^2 / pi = r^2 take the square root of both sides r = sqrt(78m^2/pi) r = 4.98 m Confidence Assessment: mostly
.............................................
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): round up my decimal to get a whole # for radius Self-critique Rating: ********************************************* Question: `q012. Summary Question 1: How do we visualize the area of a rectangle? ********************************************* Your solution: We pictured the rectangle to be made up of square units and took the formula A = l * w Confidence Assessment:
.............................................
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: ********************************************* Question: `q013. Summary Question 2: How do we visualize the area of a right triangle? ********************************************* Your solution: We pictured it as two right triangles together like a rectangle and done the formula A = ½ b*h to get the area of one triangle. Confidence Assessment: positive
.............................................
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): ok Self-critique Rating: ********************************************* Question: `q014. Summary Question 3: How do we calculate the area of a parallelogram? ********************************************* Your solution: We pictured it like a rectangle and did A = b * h Confidence Assessment: positive
.............................................
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-critique Rating: ********************************************* Question: `q015. Summary Question 4: How do we calculate the area of a trapezoid? ********************************************* Your solution: we took the formula A = w * altitude Confidence Assessment: positive
.............................................
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): ok Self-critique Rating: ********************************************* Question: `q016. Summary Question 5: How do we calculate the area of a circle? ********************************************* Your solution: we worked with the formula A = pi * r^2 to find area Confidence Assessment: positive
.............................................
Given Solution: `aWe use the formula A = pi r^2, where r is the radius of the circle. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok 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: we took the formula C = 2pi r to find the circumference. Area calls for r^2 and circumference is not in square units. Confidence Assessment: positive
.............................................
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-critique Rating: ********************************************* 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 reminded myself of formulas, how to take shapes and form them into other shapes to figure out my equation, I also learned how to take two formulas to find the solution to one. Self-critique Rating: