Download Use a proportion to find the height of the smaller cylinder. Find ... and more Summaries Dimensional Analysis in PDF only on Docsity!
Determine whether each pair of solids is similar , congruent , or neither****. If the solids are similar, state the scale factor.
SOLUTION: Ratio of radii: Ratio of heights: The ratios of the corresponding measures are equal, so the solids are similar. The scale factor is 4:3. Since the scale factor is not 1:1, the solids are not congruent. ANSWER: similar; 4:
SOLUTION: The base of one pyramid is a quadrilateral, while the base of the other is a triangle, so they are neither congruent nor similar. ANSWER: neither
- Two similar cylinders have radii of 15 inches and 6 inches. If the surface area of the larger cylinder is 2592 square inches, what is the surface area of the smaller cylinder? SOLUTION: Find the height of the larger cylinder.
Use a proportion to find the height of the smaller
cylinder.
Find the surface area of the smaller cylinder.
ANSWER:
414.7 in.^2
4. Two similar rectangular prisms have surface areas of
83.2 square feet and 20.8 square feet, respectively.
If the volume of the first prism is 46.5 cubic feet,
what is the volume of the second prism rounded to
the nearest tenth?
SOLUTION:
If two similar solids have surface areas with a ratio of a^2 : b^2 , then the volumes have a ratio of a^3 :b^3. Therefore, the ratio is 1:8 for the volumes. Use a proportion to find the volume of the smaller prism. ANSWER: 5.8 ft^3
- EXERCISE BALLS A company sells two different sizes of exercise balls. The ratio of the diameters is 15:11. If the diameter of the smaller ball is 55 centimeters, what is the volume of the larger ball? Round to the nearest tenth. SOLUTION: Since the diameter of the large ball is 75 cm, the radius is 37.5 cm. Use the formula for the volume of a sphere to find the volume of the large ball. ANSWER: 220,893.2 cm^3
SOLUTION:
All spheres are similar. Find the scale factor. Ratio of radii: The scale factor is 6:5. Since the scale factor is not 1:1, the solids are not congruent. ANSWER: similar; 6:
- Two similar pyramids have slant heights of 6 inches
and 12 inches. If the volume of the large pyramid is
548 cubic inches, what is the volume of the small
pyramid?
SOLUTION:
Find the scale factor. The scale factor is. If the scale factor is , then the ratio of volumes is .
Use a proportion to find the volume of the smaller
figure.
ANSWER:
68.5 in.^3
- Two similar cylinders have heights of 35 meters and
25 meters. The volume of the shorter cylinder is
125π cubic meters. What is the volume of the taller
cylinder?
SOLUTION:
Find the scale factor. The scale factor is. If the scale factor of the cylinders is , then the ratio of their volumes is. The ratio of the volumes is 343:125. Use the ratio to find the volume of the taller cylinder. If the shorter cylinder has a volume of 125π m^3 , then the taller cylinder has a volume of 343π m^3. ANSWER: 343π m^3
- Two spheres have surface areas of 100π square centimeters and 16π square centimeters. What is the ratio of the volume of the large sphere to the volume of the small sphere? SOLUTION:
The ratio of the surface areas is. Find the scale
factor.
Therefore, the scale factor is 5:2. If the scale factor is , then the ratio of volumes is . The ratio of the volumes is 125:8. ANSWER: 125:
- FOOD A small cylindrical can of tuna has a radius of 4 centimeters and a height of 3.8 centimeters. A larger and similar can of tuna has a radius of 5. centimeters. a. What is the scale factor of the cylinders? b. What is the volume of the larger can? Round to the nearest tenth. SOLUTION: a. Find the scale factor. b. ANSWER: a. 10: b. 419.6 cm^3
- SUITCASES Two suitcases are similar rectangular prisms. The smaller suitcase is 68 centimeters long, 47 centimeters wide, and 27 centimeters deep. The larger suitcase is 85 centimeters long. a. What is the scale factor of the prisms? b. What is the volume of the larger suitcase? Round to the nearest tenth. SOLUTION: a. Find the scale factor. b. ANSWER: a. 4: b. 168,539.1 cm^3
17. CLASS RINGS The 12-foot replica of the Aggie
Ring at Haynes Ring Plaza on the Texas A&M
campus weighs about 6500 pounds. If the ring was
based on a ring that was 0.75 inches high, what is
the scale factor?
SOLUTION:
12 feet = 144 inches The scale factor is 192:1. ANSWER:
- The pyramids shown are congruent. a. What is the perimeter of the base of Pyramid A? b. What is the area of the base of Pyramid B? c. What is the volume of Pyramid B? SOLUTION: a. Since the pyramids are congruent, the scale factor is 1:1. The base of the pyramids is in the shape of a right triangle. The base of the base of pyramid B is 8, so the same is true for pyramid A. Find the height of the base of pyramid A. The perimeter of the base of Pyramid A is 10 + 8 + 6 or 24 cm. b. From part a , we know that the bases of the pyramids are right triangles with base 8 cm and height 6 cm. The area of the base of pyramid B is or 24 cm^2. c. The area of the base of pyramid B is 24 cm^2. The height of pyramid B is the same as the height of pyramid A: 13 cm. ANSWER: a. 24 cm b. 24 cm^2 c. 104 cm^3
radius of 6 cm and a height of 7.2 cm. Volume of smaller solid = volume of the cone + volume of the cylinder So, the volume of the smaller solid is cm^3. Use the equal ratios to find the volume of the larger solid. Therefore, the volume of the larger solid is about 2439.6 cm^3. ANSWER: 2439.6 cm^3
- DIMENSIONAL ANALYSIS Two cylinders are similar. The height of the first cylinder is 23 cm and the height of the other cylinder is 8 in. If the volume of the first cylinder is 552π cm^3 , what is the volume of the other prism? Use 2.54 cm = 1 in. SOLUTION: Find the scale factor. Find the radius of the first cylinder. Now, find the radius of the second. ANSWER: 380.65π cm^3
- DIMENSIONAL ANALYSIS One spheres has a radius of 10 feet. The volume of a second sphere is 0.9 cubic meters. Use 2.54 cm = 1 in. to determine the scale factor from the first sphere to the second. SOLUTION: So, the scale factor is about 5.08 to 1. ANSWER: about 5.08 to 1
- ALGEBRA Two similar cones have volumes of 343π cubic centimeters and 512π cubic centimeters. The height of each cone is equal to 3 times its radius. Find the radius and height of both cones. SOLUTION: Let r represent the radius of the smaller cone and 3 r represent its height. Use the volume to find the value of r.
So, the radius of the smaller cone is 7 cm and its
height is 3 × 7 or 21 cm.
The ratio of the volumes equals the cube of the ratio
of the radii for the two cones.
Therefore, the radius of the larger cone is 8 cm and
its height is 3 × 8 or 24 cm.
ANSWER:
smaller cone: r = 7 cm, h = 21 cm; larger cone: r = 8 cm, h = 24 cm
c. 3.375 times as great
- CRITIQUE ARGUMENTS Cylinder X has a diameter of 20 centimeters and a height of 11 centimeters. Cylinder Y has a radius of 30 centimeters and is similar to Cylinder X. Did Laura or Paloma correctly find the height of Cylinder Y? Explain your reasoning. SOLUTION: When similar figures are compared, we need to compare corresponding parts, like this: Paloma incorrectly compared the diameter of X to the radius of Y. ANSWER: Laura; because she compared corresponding parts of the similar figures. Paloma incorrectly compared the diameter of X to the radius of Y.
- CHALLENGE The ratio of the volume of Cylinder A to the volume of Cylinder B is 1:5. Cylinder A is similar to Cylinder C with a scale factor of 1:2 and Cylinder B is similar to Cylinder D with a scale factor of 1:3. What is the ratio of the volume of Cylinder C to the volume of Cylinder D? Explain your reasoning. SOLUTION: Convert all of the given ratios to volumes. Now use these ratios to get ratio for C to D. Set one of the values equal to one. If and , then . If and , then . ANSWER: 8:135; The volume of Cylinder C is 8 times the volume of Cylinder A, and the volume of Cylinder D is 27 times the volume of Cylinder B. If the original ratio of volumes was 1 x :5 x , the new ratio is 8 x :135 x. So, the ratio of volumes is 8:135.
WRITING IN MATH Explain how the surface areas and volumes of the similar prisms are related. SOLUTION: ANSWER: Since the scale factor is 15:9 or 5:3, the ratio of the surface areas is 25:9 and the ratio of the volumes is 125:27. So, the surface area of the larger prism is or about 2.8 times the surface area of the smaller prism. The volume of the larger prism is or about 4.6 times the volume of the smaller prism.
- OPEN-ENDED Describe two nonsimilar triangular pyramids with similar bases. SOLUTION: Select some random values for one of the bases: (3, 4, 5) Multiply these by a constant (like 2) to get the values for the similar base: (6, 8, 10) These bases are in the ratio of 1:2. Select a value for the height of the first pyramid: 6 The height of the second pyramid cannot be 6 × 2 =
_ANSWER:_ Sample answer: a pyramid with a right triangle base of 3, 4, and 5 units and a height of 6 units; a pyramid with a right triangle base of 6, 8, and 10 units and a height of 6 units.
The two cones in the figure are similar. The volume of the large cone is 18,000 cubic inches. What is the volume of the small cone? A 18,000 in^3 B 83,333 in^3 C 3888 in^3 D 6480 in^3 SOLUTION: The scale factor is 20 : 12 or 5 : 3 from the large to the small cone. The ratio of the volumes should be 5^3 : 3^3 or 125 :
The volume of the larger cone is 18,000. Use a proportion to find the volume of the smaller cone. The correct choice is C. ANSWER: C
- There are two square pyramids. One square pyramid has a height of 9 centimeters and base edges measuring 20 centimeters. The second square pyramid has a height of 4.5 centimeters and base edges measuring 10 centimeters. What is the scale factor for the two square pyramids? A 1 to 1 B 2 to 1 C 2 to 2 D 3 to 1 E 3 to 2 SOLUTION: The edges of the first pyramid are all twice the length of the corresponding edges of the second pyramid, so the ratio is 2 to 1. The correct choice is B. ANSWER: B
- The two cylinders shown here are similar. The volume of the small cylinder is 16 cubic feet and the volume of the large cylinder is 54 cubic feet. Which expression represents the radius of the large cylinder, in feet? A B C D E SOLUTION: If the cylinders are similar, then their volumes have the ratio of. This makes the ratio of the radii of the large cylinder. The correct choice is C. ANSWER: C
- Two similar spheres have radii of 20π meters and 6π meters. How many times larger is the surface area of the large sphere compared to the surface area of the small sphere? Round to the nearest tenth. SOLUTION: If the ratio of the radii of the two spheres is , then the ratio of their surface areas is . ANSWER:
- MULTI-STEP Triangular prism A and triangular prism B are similar. The scale factor of prism A to prism B is 2 to 5. a. The height of the triangular base of prism A is 4. feet. Find the height of the triangular base of prism B. b. If the length of a side of prism A is 6 feet, what is the length of the corresponding side of prism B? c. If prism B has a surface area of 80 square feet, what is the surface area of prism A? d. If the volume of prism A is 32 cubic feet, what is the volume of prism B? SOLUTION: MULTI-STEP Triangular prism A and triangular prism B are similar. The scale factor of prism A to prism B is 2 to 5. a. If he height of the triangular base of prism A is 4. feet, use a proportion to find the height of the triangular base of prism B. b. If the length of a side of prism A is 6 feet, use a proportion to find the length of the corresponding side of prism B. c. If prism B has a surface area of 80 square feet, use a proportion to find the surface area of prism A. The surface areas are in ratio of 4 to 25. d. If the volume of prism A is 32 cubic feet, use a proportion to find the volume of prism B. The volumes are in a ratio of 8 to 125.
