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Determine whether each pair of solids is similar,congruent, or neither. If the solids are similar,state the scale factor.

1.

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. Sincethe scale factor is not 1:1, the solids are notcongruent.

ANSWER:

similar; 4:3

2.

SOLUTION:

The base of one pyramid is a quadrilateral, while thebase of the other is a triangle, so they are neithercongruent nor similar.

ANSWER:

neither

3. Two similar cylinders have radii of 15 inches and 6inches. If the surface area of the larger cylinder is2592 square inches, what is the surface area of thesmaller cylinder?

SOLUTION:

Find the height of the larger cylinder.

Use a proportion to find the height of the smallercylinder.

Find the surface area of the smaller cylinder.

ANSWER:

414.7 in.2

eSolutions Manual - Powered by Cognero

11-6 Volume and Nonrigid Transformations

4. Two similar rectangular prisms have surface areas of83.2 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 tothe nearest tenth?

SOLUTION:

If two similar solids have surface areas with a ratio

of a2:b2, then the volumes have a ratio of a3:b3.

Therefore, the ratio is 1:8 for the volumes. Use aproportion to find the volume of the smaller prism.

ANSWER:

5.8 ft3

5. EXERCISE BALLS A company sells two differentsizes of exercise balls. The ratio of the diameters is15:11. If the diameter of the smaller ball is 55centimeters, what is the volume of the larger ball?Round to the nearest tenth.

SOLUTION:

Since the diameter of the large ball is 75 cm, theradius is 37.5 cm.Use the formula for the volume of a sphere to findthe volume of the large ball.

ANSWER:

220,893.2 cm3



REGULARITY Determine whether each pair ofsolids is similar, congruent, or neither. If thesolids are similar, state the scale factor.

6.

SOLUTION:

Ratio of lengths:

Ratio of widths:

Ratio of heights:

The ratios of the corresponding measures are equal,so the solids are similar. The scale factor is 9:8. Sincethe scale factor is not 1:1, the solids are notcongruent.

ANSWER:

similar; 9:8

7.

SOLUTION:

Ratio of radii:

Ratio of heights:

Since the ratios of corresponding measures are notequal, the prisms are neither congruent nor similar.

ANSWER:

neither

8.

SOLUTION:

Ratio of radii:

Find the height of the second cylinder using thePythagorean Theorem.

Ratio of heights:

The ratios of the corresponding measures are equal,so the solids are similar. The scale factor is 1:1. Sincethe scale factor is 1:1, the solids are congruent.

ANSWER:

similar and congruent; 1:1



9.

SOLUTION:

All spheres are similar. Find the scale factor.

Ratio of radii:

The scale factor is 6:5. Since the scale factor is not1:1, the solids are not congruent.

ANSWER:

similar; 6:5

10. Two similar pyramids have slant heights of 6 inchesand 12 inches. If the volume of the large pyramid is548 cubic inches, what is the volume of the smallpyramid?

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 smallerfigure.

ANSWER:

68.5 in.3



11. Two similar cylinders have heights of 35 meters and25 meters. The volume of the shorter cylinder is125π cubic meters. What is the volume of the tallercylinder?

SOLUTION:


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 tofind the volume of the taller cylinder.

If the shorter cylinder has a volume of 125π m3, then

the taller cylinder has a volume of 343π m3.

ANSWER:

343π m3

12. Two spheres have surface areas of 100π squarecentimeters and 16π square centimeters. What is theratio of the volume of the large sphere to the volumeof 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:

125:8



13. Two similar hexagonal prisms have heights of 15 feetand 3 feet, respectively. If the volume of the firsthexagonal prism is 250 cubic feet, what is thevolume of the second hexagonal prism?

SOLUTION:

Therefore, the scale factor is 15:3 or 5:1, so the ratioof the volumes is 125:1.

The volume of the small cylinder is 2 ft3.

ANSWER:

2 ft3

14. ICE CREAM Two similar ice cream containershave volumes of 270π and 640π cubic inches. If theheight of the larger cylinder is 10 inches, what is thearea of the base of the smaller cylinder in terms ofπ?

SOLUTION:

volumes of 270π and 640π cubic inches. If theheight of the larger cylinder is 10 inches, what is thearea of the base of the smaller cylinder in terms ofπ? The volume of the larger cylinder is 640π cubicinches and its height is 10 inches, so the area of itsbase is 64π square inches.

If the ratio of volumes is , then the scale factor is

and the ration of the bases is .

The ratio of the volumes of the cylinders is 27:64, sothe scale factor is 3:4, and the ratio of the areas oftheir bases is 9:16 Use a proportion to find the area of the base of thesmaller cylinder.

Thus, the area of the base of the smaller cylinder is in2.

ANSWER:

36π in2



15. FOOD A small cylindrical can of tuna has a radius of4 centimeters and a height of 3.8 centimeters. Alarger and similar can of tuna has a radius of 5.2centimeters.a. What is the scale factor of the cylinders?b. What is the volume of the larger can? Round tothe nearest tenth.

SOLUTION:

a. Find the scale factor.

b.

ANSWER:

a. 10:13b. 419.6 cm3

16. SUITCASES Two suitcases are similar rectangularprisms. The smaller suitcase is 68 centimeters long,47 centimeters wide, and 27 centimeters deep. Thelarger suitcase is 85 centimeters long.a. What is the scale factor of the prisms?b. What is the volume of the larger suitcase? Roundto the nearest tenth.

SOLUTION:

a. Find the scale factor.

b.

ANSWER:

a. 4:5b. 168,539.1 cm3



17. CLASS RINGS The 12-foot replica of the AggieRing at Haynes Ring Plaza on the Texas A&Mcampus weighs about 6500 pounds. If the ring wasbased on a ring that was 0.75 inches high, what isthe scale factor?

SOLUTION:

12 feet = 144 inches

The scale factor is 192:1.

ANSWER:

192:1

18. 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 factoris 1:1. The base of the pyramids is in the shape of aright triangle. The base of the base of pyramid B is 8, so the sameis true for pyramid A.Find the height of the base of pyramid A.

The perimeter of the base of Pyramid A is 10 + 8 + 6or 24 cm. b. From part a, we know that the bases of thepyramids are right triangles with base 8 cm andheight 6 cm. The area of the base of pyramid B is or 24

cm2. c. The area of the base of pyramid B is 24 cm2. Theheight of pyramid B is the same as the height ofpyramid A: 13 cm.

ANSWER:

a. 24 cmb. 24 cm2

c. 104 cm3



19. TECHNOLOGY Jalissa and Mateo each have thesame type of digital media player, but in differentcolors. The players are congruent rectangular prisms.The volume of Jalissa’s player is 4.92 cubic inches,the width is 2.4 inches, and the depth is 0.5 inch.What is the height of Mateo’s player?

SOLUTION:

Since the rectangular prisms are congruent, the scalefactor is 1:1. Since the scale factor is 1:1, bothrectangular prisms have the same length, width, andheight.

Since the height of Jalissa’s player is 4.1 inches, theheight of Mateo’s player is 4.1 inches.

ANSWER:

4.1 in.

MODELING Each pair of solids below issimilar.

20. What is the surface area of the smaller solid shownbelow?

SOLUTION:

The ratio of the surface areas of the larger andsmaller solids equals the square of the ratio of thelengths of any of their corresponding edges.The lengths of the corresponding edges of the baseare 14 cm and 12 cm. Find the surface area of thelarger solid. The larger solid is a composite solid composed of asquare-based prism with sides of 14 cm and a heightof 12 cm and a square-based pyramid with sides of14 cm and a slant height of 15 cm.

Surface area of composite = lateral area of the prism+ area of its bottom base + lateral area of thepyramid

So, the surface area of the large solid is 1288 cm2. Use the equal ratios to find the surface area of thesmaller solid.

Therefore, the surface area of the smaller solid isabout 946.3 cm2.

ANSWER:

946.3 cm2

21. What is the volume of the larger solid shown below?

SOLUTION:

Since the solids are similar, the ratio of the volumes ofthe smaller and larger solids will equal the cube of theratio of the diameters of their circular bases.The diameter of the base of the smaller solid is 2 × 6

or 12 cm. The ratio of the diameters is .

Each solid is a composite composed of a cone atop acylinder. The volume of the smaller solid will equalthe volume of the cone with a radius of 6 cm and aheight of 5.7 cm and the volume of a cylinder with a



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 cm3. Use the equal ratios to find the volume of the largersolid.

Therefore, the volume of the larger solid is about

2439.6 cm3.

ANSWER:

2439.6 cm3

22. DIMENSIONAL ANALYSIS Two cylinders aresimilar. The height of the first cylinder is 23 cm andthe height of the other cylinder is 8 in. If the volume

of the first cylinder is 552π cm3, what is the volumeof the other prism? Use 2.54 cm = 1 in.

SOLUTION:


Find the radius of the first cylinder.

Now, find the radius of the second.

ANSWER:

380.65π cm3



23. DIMENSIONAL ANALYSIS One spheres has aradius of 10 feet. The volume of a second sphere is0.9 cubic meters. Use 2.54 cm = 1 in. to determinethe 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

24. ALGEBRA Two similar cones have volumes of343π 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 3rrepresent its height. Use the volume to find the valueof r.

So, the radius of the smaller cone is 7 cm and itsheight is 3 × 7 or 21 cm. The ratio of the volumes equals the cube of the ratioof the radii for the two cones.

. Therefore, the radius of the larger cone is 8 cm andits height is 3 × 8 or 24 cm.

ANSWER:

smaller cone: r = 7 cm, h = 21 cm; larger cone: r = 8cm, h = 24 cm



25. TENTS Two tents are in the shape of hemispheres,with circular floors. The ratio of their floor areas is9:12.25. If the diameter of the smaller tent is 6 feet,what is the volume of the larger tent? Round to thenearest tenth.

SOLUTION:

Therefore, the ratio of the scale factor is 3:3.5.Find the radius of the large tent.

ANSWER:

89.8 ft3

26. MODELING In this problem, you will investigatesimilarity. The heights of two similar cylinders are inthe ratio 2 to 3. The lateral area of the larger cylinderis 162π square centimeters, and the diameter of thesmaller cylinder is 8 centimeters.a. VERBAL What is the height of the largercylinder? Explain your method.b. GEOMETRIC Sketch and label the twocylinders.c. ANALYTICAL How many times as great is thevolume of the larger cylinder as the volume of thesmaller cylinder?

SOLUTION:

a. Find the diameter of the larger cylinder.

The diameter is 12, so the radius is 6. Use this andthe lateral area to find the height of the largercylinder.

.b. Sample answer:

c. Simplify the ratio of the volume of the largercylinder to the volume of the smaller cylinder.

ANSWER:

a. 13.5 cm; The heights are in the ratio 2 to 3, so the

scale factor is 2:3. Since , the diameter of the

larger cylinder is 12 cm. The lateral area of the larger

cylinder is 162π cm2 , so the height is 162π ÷ 12π or13.5 cm.b. Sample answer:



c. 3.375 times as great

27. CRITIQUE ARGUMENTS Cylinder X has adiameter of 20 centimeters and a height of 11centimeters. Cylinder Y has a radius of 30centimeters and is similar to Cylinder X. Did Laura orPaloma correctly find the height of Cylinder Y?Explain your reasoning.

SOLUTION:

When similar figures are compared, we need tocompare corresponding parts, like this:

Paloma incorrectly compared the diameter of X tothe radius of Y.

ANSWER:

Laura; because she compared corresponding parts ofthe similar figures. Paloma incorrectly compared thediameter of X to the radius of Y.

28. CHALLENGE The ratio of the volume of CylinderA to the volume of Cylinder B is 1:5. Cylinder A issimilar to Cylinder C with a scale factor of 1:2 andCylinder B is similar to Cylinder D with a scale factorof 1:3. What is the ratio of the volume of Cylinder Cto 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 oneof the values equal to one.

If and , then

.

If and , then

.

ANSWER:

8:135; The volume of Cylinder C is 8 times thevolume of Cylinder A, and the volume of Cylinder Dis 27 times the volume of Cylinder B. If the originalratio of volumes was 1x :5x, the new ratio is 8x :135x.So, the ratio of volumes is 8:135.



29. WRITING IN MATH Explain how the surfaceareas and volumes of the similar prisms are related.

SOLUTION:

ANSWER:

Since the scale factor is 15:9 or 5:3, the ratio of thesurface 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.

30. OPEN-ENDED Describe two nonsimilar triangularpyramids 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 valuesfor the similar base: (6, 8, 10) These bases are in the ratio of 1:2. Select a value forthe height of the first pyramid: 6 The height of the second pyramid cannot be 6 × 2 =12.

ANSWER:

Sample answer: a pyramid with a right triangle baseof 3, 4, and 5 units and a height of 6 units; a pyramidwith a right triangle base of 6, 8, and 10 units and aheight of 6 units.



31. PERSEVERANCE Plane P is parallel to the base ofcone C, and the volume of the cone above the plane

is of the volume of cone C. Find the height of cone

C.

SOLUTION:

Since plane P is parallel to the base of cone C, thecone above the plane is similar to cone C. The ratioof the volumes of the cones will equal the cube of theratio of their heights.

Therefore, the height of cone C is 14 cm.

ANSWER:

14 cm

32. WRITING IN MATH Explain why all spheres aresimilar.

SOLUTION:

Sample answer: All spheres are the same shape. Theonly parameter that can vary is the radius, so allspheres are similar.

ANSWER:

Sample answer: All spheres are the same shape. Theonly parameter that can vary is the radius, so allspheres are similar.

33. Two similar rectangular prisms have volumes of 60cubic meters and 480 cubic meters. The surface areaof the smaller prism is 94 square meters. What is thesurface area of the larger prism?

A 188 m²B 376 m²C 514 m²D 752 m²

SOLUTION: The ratio of the volumes of the two prisms is

. This makes the scale factor .

The ratio of the surface areas of the two prisms is

. Set up a proportion using the ratio of the squareof the scale factor.

The correct choice is B.

ANSWER: B

34. Two spherical beach balls have diameters of 8 inchesand 20 inches. Which of the following statementsmust be true?

A The ratio of the radii is 2 : 5.B The ratio of the surface areas is 8 : 125.C The ratio of the volumes is 4 : 25.D The ratio of the circumferences is 4 : 25.

SOLUTION: Use the given diameters to find the ratio of the balls'radii.

Therefore, the ratio of the balls' radii is 2:5. Thecorrect choice is A.

ANSWER: A



35. The two cones in the figure are similar. The volumeof the large cone is 18,000 cubic inches. What is thevolume of the small cone?

A 18,000 in3

B 83,333 in3

C 3888 in3

D 6480 in3

SOLUTION: The scale factor is 20 : 12 or 5 : 3 from the large tothe small cone.

The ratio of the volumes should be 53 : 33 or 125 :27. The volume of the larger cone is 18,000. Use aproportion to find the volume of the smaller cone.

The correct choice is C.

ANSWER: C

36. There are two square pyramids. One square pyramidhas a height of 9 centimeters and base edgesmeasuring 20 centimeters. The second squarepyramid has a height of 4.5 centimeters and baseedges measuring 10 centimeters. What is the scalefactor for the two square pyramids?

A 1 to 1B 2 to 1C 2 to 2D 3 to 1E 3 to 2

SOLUTION: The edges of the first pyramid are all twice the lengthof the corresponding edges of the second pyramid, sothe ratio is 2 to 1. The correct choice is B.

ANSWER: B



37. The two cylinders shown here are similar. Thevolume of the small cylinder is 16 cubic feet and thevolume of the large cylinder is 54 cubic feet.

Which expression represents the radius of the largecylinder, in feet?

A

B

C

D

E

SOLUTION: If the cylinders are similar, then their volumes havethe ratio of . This makes the ratio of the

radii of the large cylinder . The correct

choice is C.

ANSWER: C

38. Two similar spheres have radii of 20π meters and 6πmeters. How many times larger is the surface area ofthe large sphere compared to the surface area of thesmall 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: 11.1

39. MULTI-STEP Triangular prism A and triangularprism 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 4.8feet. Find the height of the triangular base of prism B.

b. If the length of a side of prism A is 6 feet, what isthe 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 isthe volume of prism B?

SOLUTION: MULTI-STEP Triangular prism A and triangularprism B are similar. The scale factor of prism A toprism B is 2 to 5.

a. If he height of the triangular base of prism A is 4.8feet, use a proportion to find the height of thetriangular base of prism B.

b. If the length of a side of prism A is 6 feet, use aproportion to find the length of the corresponding sideof 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 aproportion to find the volume of prism B. Thevolumes are in a ratio of 8 to 125.



ANSWER: a. 12 ftb. 15 ft

c. 12.8 ft2

d. 500 ft3



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