Area, Perimeter, Volume and Pythagorean Theorem Assessment Name: 1. Find the perimeter of a right triangle with legs measuring 10 inches and 24 inches a. 34 inches b. 60 inches c. 120 inches d. 240 inches 2. The side lengths of the base of right prism are doubled while the height is not changed. Which of the following best describes the result on the volume of the prism a. The volume remains the same b. The volume doubles c. The volume triples d. The volume quadruples 3. A square pasture is bordered on one side by a stream and on the other three sides by a fence. If the fence is 204 feet long, what would be the area of the pasture? a. 408 square feet b. 2,601 square feet c. 4,624 square feet d. 10,404 square feet Designing Boxes Assessment Page 1 of 13
4. A cylinder has radius of 6 cm and height of 8 cm what would be its volume? a. 48 cubic cm b. 96 cubic cm c. 226 cubic cm d. 905 cubic cm 5. A cylindrical tank has diameter of 6 m and height of 10 m what would be its surface area? a. 60 square m b. 90 square m c. 245 square cm d. 528 square cm 6. If the volume of a new container is 8 times larger than a previous container, by how much has each dimension increased? a. 2 times b. 3 times c. 4 times d. 8 times Designing Boxes Assessment Page 2 of 13
7. Because of a change in her company s best selling product, Korie has been given the job of redesigning the packaging for the latest product upgrade. The marketing department has told her that the package needs to be larger to catch the customers eye on the shelf. She decides to double each edge length. The volume of the old container was 27 cubic inches. What is the volume of the new package? The volume of the new package is cubic inches. Designing Boxes Assessment Page 3 of 13
8. Harold is the marketing manager for a major toy company. His company is ready to release a new toy. Harold s research tells him that more people will buy the toy if the front of the box has a surface area of 54 square inches. The depth of the box Harold designs is the same as one of the edge lengths of the front of the box. All of Harold s measurements are whole numbers (no fractions or decimals). As always, Harold designs a box that sells well. What is the volume of the box he designs? The volume of the new package is cubic inches. Designing Boxes Assessment Page 4 of 13
9. Jerry works for a shipping company whose customer has requested a square based container with a volume of 96 cubic feet. The dimensions of the container must be whole number values (no fractions or decimals) per the customer s request. As always, Jerry makes his customer happy. What are the dimensions of the container he designs? The dimensions of the container are feet. Designing Boxes Assessment Page 5 of 13
10. As the product manager for a packaging company, Chuck is responsible for creating the package that best fits his customers needs. His customer has decided to change their packaging for the summer season. The current package holds a pen that reaches diagonally through the center of a cube measuring 8 inches on an edge. Chuck is given the job of creating a cylindrical prism container that has equal diameter and height that will hold the same pen. Chuck also needs to calculate and compare the cost of the package options. The material used to build the cube cost $0.35 per square inch and the material for the cylinder costs $0.32 per square inch. What is the diameter of the cylinder? (round your answer to two decimal places) What is the height of the cylinder? (round your answer to two decimal places) Identify the more expensive option and the cost to build it. Determine which package costs more to build and state its cost. Helpful ideas: Circumference of a circle: C = πd Area of a circle: A = πr 2 Designing Boxes Assessment Page 6 of 13
The diameter and height of the cylinder is inches. The is more expensive to build. It costs Designing Boxes Assessment Page 7 of 13
Area, Perimeter, Volume and Pythagorean Theorem Assessment Solutions Area, Perimeter, Volume and Pythagorean Theorem Quiz 3. Find the perimeter of a right triangle with legs measuring 10 inches and 24 inches a. 34 inches b. 60 inches c. 120 inches d. 240 inches 4. The side lengths of the base of right prism are doubled while the height is not changed. Which of the following best describes the result on the volume of the prism a. The volume remains the same b. The volume doubles c. The volume triples d. The volume quadruples 3. A square pasture is bordered on one side by a stream and on the other three sides by a fence. If the fence is 204 feet long, what would be the area of the pasture? a. 408 square feet b. 2,601 square feet c. 4,624 square feet d. 10,404 square feet 4. A cylinder has radius of 6 cm and height of 8 cm what is its volume? a. 48 cubic cm b. 96 cubic cm c. 226 cubic cm d. 905 cubic cm 5. A cylindrical tank has diameter of 6 m and height of 10 m what is its surface area? a. 60 square m b. 90 square m c. 245 square m d. 528 square m 6. If the volume of a new container is 8 times larger than a previous container, by how much has each dimension increased? a. 2 times b. 3 times c. 4 times d. 8 times Designing Boxes Assessment Page 8 of 13
9. Because of a change in her company s best selling product, Korie has been given the job of redesigning the packaging for the latest product upgrade. The marketing department has told her that the package needs to be larger to catch the customers eye on the shelf. She decides to double each edge length. The volume of the old container was 27 cubic inches. What is the volume of the new package? Volume was 27 cubic inches so possible container would be 3X3X3. If she doubles them then it would be 6X6X6 for a total of 216 cubic inches. OR As edge lengths all increase by a factor of 2, the volume increases by 2 3 So, new volume is 8X27 = 216 cubic inches. The volume of the new package is 216 cubic inches. Designing Boxes Assessment Page 9 of 13
10. Harold is the marketing manager for a major toy company. His company is ready to release a new toy. Harold s research tells him that more people will buy the toy if the front of the box has a surface area of 54 cubic inches. The depth of the box Harold designs is the same as one of the edge lengths of the front of the box. All of Harold s measurements are whole numbers (no fractions or decimals). As always, Harold designs a box that sells well. What is the surface area of the box he designs? Surface area is 54 square inches and two of the dimensions are equal, so I need a perfect square that divides 54. That number is 9. Thus, the equal edges are 3 inches each and the third is 54 divided by 9 or 6. So the box is 3X3X6 and its surface area is: Sides: 3X6X4 = 72 square inches Top and Bottom: 3X3X2 = 18 square inches Then the total surface area is 90 square inches The surface area of the new package is 90 square inches. Designing Boxes Assessment Page 10 of 13
11. Jerry works for a shipping company whose customer has requested a square based container with a volume of 96 cubic feet. The dimensions of the container must be whole number values (no fractions or decimals) per the customer s request. As always, Jerry makes his customer happy. What are the dimensions of the container he designs? I need a perfect square that evenly divides 96. That number is 16. So two of the dimensions are 4 feet and 4 feet. Then the third dimension is 96 divided by 16 or 6. The dimensions of the container are 4X4X6 feet. Designing Boxes Assessment Page 11 of 13
10. As the product manager for a packaging company, Chuck is responsible for creating the package that best fits his customers needs. His customer has decided to change their packaging for the summer season. The current package holds a pen that reaches diagonally through the center of a cube measuring 8 inches on an edge. Chuck is given the job of creating a cylindrical prism container that has equal diameter and height that will, hold the same pen. Chuck also needs to calculate and compare the cost of the package options. The material used to build the cube cost $0.35 per square inch and the material for the cylinder costs $0.32 per square inch. What is the diameter of the cylinder? (Round your answer to two decimal places) What is the of the cylinder? (Round your answer to two decimal places) Identify the more expensive option and the cost to build it. Helpful ideas: Circumference of a circle: C = πd Area of a circle: A = πr 2 Show work to support your answers using words, numbers and/or diagrams. Designing Boxes Assessment Page 12 of 13
First, find diagonal of cube: 8sqrt3 13.86 Then in the cylinder the diagonal is 13.86 and since diameter and height are equal, use Pythagorean Theorem to solve for diameter and height with equal leg lengths (alternately use 45-45-90 relationship). Thus we get dimension of cylinder of 9.80 inches diameter and height Then find S.A. of cube: 64x6 = 384 square inches. Use formulas on front to find S.A. of cylinder 9.8 2 (π)+ 2(π)4.9 2 = 452 square inches Now cost to produce Cube is 384x0.35 = $134.40 Cost to produce Cylinder is 452x0.32 = $144.64 So the cylinder costs more to produce. The diameter and height of the cylinder is 9.80 inches. The cylinder is more expensive to build. It costs $144.64 Designing Boxes Assessment Page 13 of 13