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Home , Pentagonal prism, Prism (geometry), Pyramid (geometry), Surface area, Triangular prism

1-8 Three-Dimensional Figures

Determine whether the solid is a . Find the and of each solid Then identify the solid. If it is a polyhedron, to the nearest tenth. name the bases, faces, edges, and vertices.

3. 1. SOLUTION: SOLUTION: The formulas for finding the volume and surface A polyhedron is a solid made from flat surfaces that of a are and , enclose a single region of space. This solid has where S = total , V = volume, h = height curved surfaces, so it is not a polyhedron. The given of a solid, B = area of the , P = of the figure is a solid with congruent parallel circular bases base. connected by a curved surface. Therefore, it is a . Since the base of the prism is a , the perimeter P of the base is or 14 ANSWER: centimeters. The area of the base B is not a polyhedron; cylinder or 12 centimeters. The height is 3 centimeters.

2.

SOLUTION: The solid is formed by polygonal faces, so it is a polyhedron. It has a rectangular base and three or The surface area of the prism is 66 square more triangular faces that meet at a common . centimeters. So, it is a rectangular . Base: Faces: Edges: Vertices, K, L, M, N, J The volume of the prism is 36 cubic centimeters. ANSWER: a polyhedron; rectangular pyramid; base: ANSWER: faces edges: 66 cm2 ; 36 cm3 vertices: K, L, M, N, J

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5. CUPCAKES LaMea is icing cupcakes with a -shaped icing bag 3.5 inches in , 5 inches tall, with a slant height of approximately 5.3 4. inches. The icing bag has no top. Find each measure SOLUTION: to the nearest tenth. a. the volume of icing that will fill the bag The formulas for finding the volume and surface area b. the area of plastic used to make the icing bag of a are and , where S SOLUTION: = total surface area, V = volume, and r = radius. a. The formula for the volume of a cone is Here, in. where r is the radius of the base and h is the height of the cone. Since the diameter of the cone base is 3.5 inches, the radius is 1.75 inches.

The volume of the sphere is or about 904.8 The volume of the icing bag is about 16 cubic inches. cubic inches.

b. To find the area of the plastic used to make the icing bag, find the lateral area of the cone. This is the surface area minus the area of the base or , where S = total surface area, V = volume, r = radius, = slant height, and h = height. The surface area of the sphere is or about 452.4 square inches.

ANSWER: 144π or about 452.4 in2 ; 288π or about 904.8 in3 The area of the plastic used to make the icing bag is approximately 29.1 square inches.

ANSWER: a. ≈16.0 in 3 b. ≈29.1 in2

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Identify the solid modeled by each object. State 9. Refer to Page 80. whether the solid modeled is a polyhedron. SOLUTION: 6. Refer to Page 80. This object models a solid that has two parallel SOLUTION: congruent rectangular bases connected by four This object models a solid with a circular base rectangular faces, so it is a rectangular prism. The connected by a curved surface to a single vertex. So solid is formed by polygonal faces, so it is a it is a cone. A solid with all flat surfaces that enclose polyhedron. a single region of space is called a polyhedron. This solid has a curved surface, so it is not a polyhedron. ANSWER: rectangular prism; a polyhedron ANSWER: cone; not a polyhedron 10. Refer to Page 80. SOLUTION: 7. Refer to Page 80. This object models a solid that is a set of points in space that are the same distance from a given . SOLUTION: So, it is a sphere. A sphere has no faces, edges, or This object models a solid that has two visible vertices, so it is not a polyhedron. triangular faces that meet at a common vertex. So, it is a pyramid. The type of pyramid will be determined ANSWER: by its base, which is not visible. The base will also sphere; not a polyhedron determine the total number of triangular faces. The solid is formed by polygonal faces, so it is a 11. Refer to Page 80. polyhedron. SOLUTION: ANSWER: This object models a solid with congruent parallel pyramid; a polyhedron circular bases connected by a curved surface. Therefore, it is a cylinder. This solid has a curved 8. Refer to Page 80. surface, so it is not a polyhedron.

SOLUTION: ANSWER: This object models a solid has parallel triangular cylinder; not a polyhedron bases connected by three rectangular faces. So, it is a . It is formed by polygonal faces, so it is a polyhedron.

ANSWER: triangular prism; a polyhedron

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STRUCTURE Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices. 13.

SOLUTION: A solid with all flat surfaces that enclose a single 12. region of space is called a polyhedron. This solid has a curved surface, so it is not a polyhedron. The given SOLUTION: figure is a solid with a circular base connected by a The solid is formed by polygonal faces, so it is a curved surface to a single vertex. So it is a cone. polyhedron. This solid has two congruent pentagonal bases, so it is a . ANSWER: : Each flat surface is called face. not a polyhedron; cone Edges: The segments where the faces intersect are called edges. Vertex: The point where three or more edges intersect is called a vertex. Bases: ABCDE, FGHJK Faces: ABCDE, FGHJK 14. SOLUTION: Edges: This solid is formed by polygonal faces, so it is a Vertices: polyhedron. It has triangular bases. So, it is a ANSWER: triangular prism. Face: Each flat surface is called face. a polyhedron; pentagonal prism; bases: ABCDE, Edges: The line segments where the faces intersect FGHJK; faces: ABCDE, FGHJK; are called edges. edges: Vertex: The point where three or more edges intersect is called a vertex. vertices: A, B, C, D, E, F, G, H, J, K Bases: Faces: Edges: Vertices:

ANSWER: a polyhedron; triangular prism; bases: faces: edges: ; vertices: M, P, L, J, N, K

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15. 17.

SOLUTION: SOLUTION: This solid has no faces, edges, or vertices, so it is not The solid is formed by polygonal faces, so it is a a polyhedron. It is a set of points in space that are the polyhedron. The given pyramid has a pentagonal same distance from a given point. So, it is a sphere. base, so it is a . Faces: Each flat surface is called face. ANSWER: Edges: The line segments where the faces intersect not a polyhedron; sphere are called edges. Vertex: The point where three or more edges intersect is called a vertex. 16. Base: JHGFD Faces: SOLUTION: Edges: A solid with all flat surfaces that enclose a single Vertices: region of space is called a polyhedron. The solid has a curved surface, so it is not a polyhedron. The given ANSWER: figure is a solid with congruent parallel circular bases a polyhedron; pentagonal pyramid; base: JHGFD; connected by a curved surface. Therefore, it is a faces: JHGFD, cylinder. edges: vertices: ANSWER: J, H, G, F, D, E not a polyhedron; cylinder

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Find the surface area and volume of each solid to the nearest tenth.

19.

SOLUTION: 18. The formulas for finding the volume and surface area SOLUTION: of a prism are and , where S The formulas for finding the volume and surface area = total surface area, V = volume, h = height of a of of a prism are and , where solid, B = area of the base, and P = perimeter of the S = total surface area, V = volume, h = height of a base. solid, B = area of the base, and P = perimeter of the base. Since the base of the prism is a square, the perimeter P of the base is or 18 meters. The area of the Since the base of the prism is a rectangle, the base B is perimeter P of the base is or 14 inches. or 20.25 square meters. The height is 4.5 The area of the base B is or 10 meters. square inches. The height is 6 inches.

The surface area of the prism is 121.5 square meters.

The surface area of the prism is 104 in2.

The volume of the prism is 91.1 cubic meters.

The volume of the prism is 60 in3. ANSWER: 121.5 m2 ; 91.1 m3 ANSWER: 104 in2 ; 60 in3

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20. 21.

SOLUTION: SOLUTION: The formulas for finding the volume and surface area The formulas for finding the volume and surface area of a cone are and , of a prism are and , where S = total surface area, V = volume, h = height, B = area where S = total surface area, V = volume, r = radius, of the base, and P = perimeter of the base. = slant height, and h = height. Since the base of the prism is a , the perimeter P of the base is or 24 centimeters. The Here, the diameter of the cone is 10 yards, so the radius is 5 yards. yards and yards area of the base B is or 24 square centimeters. The height of the prism is 5 centimeters.

The surface area of the cone is or about 282.7 The surface area of the triangular prism is 168 square square yards. centimeters.

The volume of the prism is 120 cubic centimeters.

ANSWER: 168 cm2 ; 120 cm3

The volume of the cone is about 314.2 cubic yards.

ANSWER: 90π or about 282.7 yd2 ; 100π or about 314.2 yd3

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22. 23. SOLUTION: SOLUTION: The formulas for finding the volume and total surface The formulas for finding the volume and surface area of a pyramid are and , area of a cylinder are and where S = total surface area, V = volume, h = height, , where S = total surface area, V B = area of the base, P = perimeter of the base, and = volume, r = radius, and h = height. = slant height. Here, mm and mm. Since the base of the pyramid is a square, the perimeter P of the base is or 64 feet. The area of the base B is or 256 square feet.

Here, ft and ft.

The surface area of the cylinder is or about 471.2 mm2.

The surface area of the triangular prism is 800 square feet.

The volume of the cylinder is or about 785.4 mm 3.

ANSWER: 2 3 150π or about 471.2 mm ; 250π or about 785.4 mm The volume of the prism is 1280 cubic feet.

ANSWER: 800 ft2 ; 1280 ft3

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24. CARDS A rectangular card box is is 2.5 inches by 25. DRUMS Drum shell size is important to the tone of 3.5 inches. The depth of the box is 0.75 inch but the the drum. The bigger the diameter, the deeper the sound will be. Shaun’s drum has a diameter of 14 depth of cards in the box is only the depth of the inches. Suppose the height of the drum shell is 8.5 box. Find each measure to the nearest tenth. inches. Find each measure to the nearest tenth. a. the surface area of the card box [Refer to page 81] b. the volume of sand in the card box a. the volume of air within the drum SOLUTION: b. the surface area of the drum a. The formula for the surface area of a prism is SOLUTION: , where S = total surface area, h = The formulas for finding the volume and surface area height, B = area of the base, and P = perimeter of the base. of a cylinder are and , where S = total surface area, V = volume, r = radius, Since the base of the prism is a rectangle, the and h = height. perimeter P of the base is or 12 in. a. The diameter of the drum is 14 in, so the radius is 7 in. The height 8.5 in. The area of the base B is or 8.75 in2. The

height is 0.75 in.

The surface area of the box is 26.5 in2. The volume of the air in the drum is about 1308.5 in 3. b. The formula for the volume of a prism is . The depth of the card box is 0.75 inch and only half b. of the box is filled. To determine the height of the filled portion multiply 0.75 by 0.5 to get 0.375 inch. Substitute this for the height in the volume formula.

The surface area of the drum is about 681.7 in2. The volume of the cards in the box is approximately 3.3 in3. ANSWER: a. 1308.5 in3 ANSWER: b. 681.7 in 2 a.26.5 in2 b. 3.3 in 3

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26. SENSE-MAKING Bento boxes are Japanese 27. ALGEBRA The surface area of a is 54 square style lunch boxes in which several different foods are inches. Find the length of each . packed for lunch in varying compartments. The box SOLUTION: shown can be modeled by a square prism. Assuming that the layers are filled to the top, what There are six congruent sides in a cube. Each side is volume of food can this Bento box hold? in the of a square. To find the surface area of the cube, find the sum of the area of each side of a cube.

Let a be the length of each side of a cube. So, the surface area of the cube is .

SOLUTION: The height of the box is 2 + 1.5 + 1.25 or 4.75 in. The formula for finding the volume of a prism is

. The length must be positive. So,

The length of each edge is 3 inches.

ANSWER: 3 in.

The volume of the lunch box is 76 cubic inches.

ANSWER: 76 in3

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28. ALGEBRA The volume of a cube is 729 cubic 29. PAINTING Tara is painting her family’s fence. Each centimeters. Find the length of each edge. post is composed of a square prism and a . The slant height of the pyramid is 4 inches. SOLUTION: Determine the surface area and volume of each post. The formula for finding the volume of the prism is .

The base of the cube is a square, so the area of the base is . The length of height is equal to the length of the side, since all the sides are congruent in a cube.

SOLUTION: Surface area of the post is equal to the surface area of the square pyramid plus the surface area of the rectangular prism. The length of each edge is 9 cm.

ANSWER: 9 cm

ANSWER: 1200 in2 ; 1776 in3

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30. COLLECT DATA Use a ruler or tape measure and inches high, calculate the area of the cake that will be what you have learned in this lesson to find the frosted assuming there is no frosting between layers. surface area and volume of a soup can. b. Calculate the area of the cylindrical cake that will be frosted, if each layer is 4 inches in height. SOLUTION: c. If one can of frosting will cover 50 square inches A can of soup may be 3 inches in diameter and 4 of cake, how many cans of frosting will be needed inches high. Use a radius of 1.5 inches and the for each cake? formulas to find the surface area and volume of the d. If the height of each layer of cake is 5 inches, can. what does the radius of the cylindrical cake need to be, so the same amount of frosting is used for both cakes? Explain your reasoning.

SOLUTION: a. The formula for finding the surface area of a prism is , where S = total surface area, h = height, B = area of the base, and P = perimeter of the

base

Since the base of the prism is a rectangle, the perimeter P of the base is or 14 inches. The area of the base B is 4 × 3 or 12 square inches. Each cake is 3 inches high. So, the height is 6 inches. This can would have a surface area of about 51.84 in 2 and a volume of 28.27 in3. The top is not going to be frosted. So, the area to be See students' work as measurements for soup cans frosted is given by . will vary. Substitute. ANSWER: See students’ work.

31. CAKES Cakes come in many and sizes. Often they are stacked in two or more layers, like those in the diagrams shown below. The area of the cake to be frosted is 96 in2.

b. The formula for finding the surface area of a cylinder is , where S = total surface area, r = radius, and h = height.

Here, r = 2. The height of each cylindrical cake is 4 in. So, the total height is 8 in. Since the top is not going to be frosted, the area to be frosted is given by . a. If each layer of the rectangular prism cake is 3 eSolutions Manual - Powered by Cognero Page 12 1-8 Three-Dimensional Figures

area of the rectangular cake is 152 in2. To find the radius of a cylindrical cake with the same height, solve the equation 152 = πr2 + 20πr. The solutions are r = –22.18 or r = 2.18. Using a radius of 2.18 in. gives surface area of about 152 in2.

32. CHANGING UNITS A gift box has a surface area The area of the cylindrical cake to be frosted is about of 6.25 square feet. What is the surface area of the 113.1 in2. box in square inches?

c. Divide the area to be frosted by 50. SOLUTION: Surface area of the gift box = 6.25 ft2 1 foot = 12 inches So, 2 cans of frosting are needed for the rectangular Surface area of the gift box = 6.25(12 inches)2 prism cake. in2 = 900 in2 So, 3 cans of frosting are needed for the cylindrical ANSWER: cake. 2 900 in d. Find the surface area of the rectangular cake if 33. CHANGING UNITS A square pyramid has a the height of the each layer 5 in. volume of 4320 cubic inches. What is the volume of this pyramid in cubic feet?

SOLUTION: Volume of the pyramid = 4320 in3 1 foot = 12 inches The surface area of the rectangular cake is 152 in2. So, 1 inch = foot.

Volume of a pyramid To find the radius of a cylindrical cake with the same 2 height, solve the equation 152 = πr + 20πr. = Solving the equation using the quadratic formula gives r = –22.18 and r = 2.18. ft3

Since the radius can never be negative, r = 2.18. = 2.5 ft3

The same amount of frosting will be needed if the ANSWER: radius of the cake is 2.18 in. 2.5 ft3

ANSWER: a. 96 in2 b. 113.1 in2 c. prism: 2 cans; cylinder: 3 cans d. 2.18 in.; if the height is 10 in., then the surface eSolutions Manual - Powered by Cognero Page 13 1-8 Three-Dimensional Figures

34. EULER’S FORMULA The number of faces F, 35. CHANGING A rectangular prism vertices V, and edges E of a polyhedron are related has a length of 12 centimeters, width of 18 by Euler’s (OY luhrz) Formula: F + V = E + 2. centimeters, and height of 22 centimeters. Describe Determine whether Euler’s Formula is true for each the effect on the volume of a rectangular prism when of the figures in Exercises 18–23. each is doubled.

SOLUTION: SOLUTION: Use Euler’s formula: F + V = E + 2 The formula for finding the volume of a prism is Exercise 18: , where V = volume, h = height, and B = area Rectangular Prism: 6 + 8 = 12 + 2 of the base. Since the base of the prism is a So, Euler’s formula is true. The answer is “Yes”. rectangle, the area of the base B is or 216 square centimeters. Here, height of the prism = 22 Exercise 19: cm. Square Prism: 6 + 8 = 12 + 2 So, Euler’s formula is true. The answer is “Yes”.

Exercise 20: This figure isa cone and not a polyhedron, so Euler’s The volume of the original prism is 4752 cm3. Formula does not apply. So, the answer is “No”.

Double the dimensions and find the volume. Exercise 21:

Triangular Prism: 5 + 6 = 9 + 2 Volume of the new prism So, Euler’s formula is true. The answer is “Yes”.

Exercise 22: Square Pyramid: 5 +5 = 8 +2 So, Euler’s formula is true. The answer is “Yes”.

Exercise 23: This figure is a cylinder and not a polyhedron, so The volume increased by a factor of 8 when each Euler’s Formula does not apply. So, the answer is dimension was doubled. “No”. ANSWER: ANSWER: The volume of the original prism is 4752 cm3. The Exercise 18: yes, 6 + 8 = 12 + 2; Exercise 19: yes, 6 volume of the new prism is 38,016 cm3. The volume + 8 = 12 + 2;Exercise 20: no, this figure is not a increased by a factor of 8 when each dimension was polyhedron, so Euler’s Formula does not apply; doubled. Exercise 21: yes, 5 + 6 = 9 + 2; Exercise 22: yes, 5 + 5 = 8 + 2; Exercise 23: no, this figure is not a 36. MULTIPLE REPRESENTATIONS In this polyhedron, so Euler’s Formula does not apply. problem, you will investigate how changing the length of the radius of a cone affects the cone’s volume. a. TABULAR Create a table showing the volume of a cone when doubling the radius. Use radius values between 1 and 8. b. GRAPHICAL Use the values from your table to eSolutions Manual - Powered by Cognero Page 14 1-8 Three-Dimensional Figures

create a graph of radius versus volume. c. VERBAL Make a conjecture about the effect of doubling the radius of a cone on the volume. Explain your reasoning. d. ALGEBRAIC If r is the radius of a cone, write an expression showing the effect doubling the radius has on the cone’s volume.

The volume of a cone is increased by a factor of 4 when the radius is doubled.

d. The formula for finding the volume of a cone is SOLUTION: . a. The formula for finding the volume of a cone is , where V = volume, r = radius, and h = Double the radius, that is, 2r. height.

Choose radius values between 1 and 8 and find the volume of the cone, then tabulate the results. ANSWER: a.

b. Plot the points and draw a curve through the points b. on the coordinate .

c.

c. Double the radius values and calculate the volume of the cone, then tabulate the results. eSolutions Manual - Powered by Cognero Page 15 1-8 Three-Dimensional Figures

Here, height of the prism = 3 ft.

The total surface area of the prism is 94 in2. Doubling the radius results in an increase in the So, both answers are incorrect. volume by a factor of 4. ANSWER: d. Neither; sample answer: the surface area is twice the sum of the areas of the top, front, and left side of the 37. CRITIQUE Alex and Emily are calculating the prism or 2(5 · 3 + 5 · 4 + 3 · 4), which is 94 in2. surface area of the rectangular prism shown. Is either of them correct? Explain your reasoning. 38. REASONING Is a cube a ? Explain.

SOLUTION: In a cube, all of the faces are regular congruent and all of the edges are congruent. So, it is a regular polyhedron. The answer is “Yes”.

ANSWER: Yes; all of the faces are regular congruent squares and all of the edges are congruent.

SOLUTION: Sample answer: The formula for finding the surface area of a prism is , where S = total surface area, h = height, B = area of the base, and P = perimeter of the base.

Since the base of the prism is a rectangle, the perimeter P of the base is or 18 inches. The area of the base B is or 20 square inches.

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39. CHALLENGE Describe the solid that results if the 40. OPEN-ENDED Draw an irregular 14-sided number of sides of each base increases infinitely. The polyhedron which has two congruent bases. bases of each solid are regular inscribed in a SOLUTION: . a. pyramid Sample answer: If the bases are congruent are b. prism congruent, there must be 12 faces connecting the bases to make a total of 14. Therefore, the two bases SOLUTION: must be congruent 12-sided polygons. a. A pyramid is a polyhedron that has a polygonal base and three or more triangular faces that meet at a common vertex. As the number of sides for the base increases towards infinity, the for the base will approach the shape of the circle in which it is inscribed, and the triangular faces will become more of a curved surface. A cone is a solid with a ANSWER: circular base connected by a curved surface to a Sample answer: single vertex. So, as the number of sides of the base increases infinitely, the solid becomes a cone. b. A prism is a polyhedron that has two parallel congruent polygonal bases connected by faces. As the number of sides for the bases increases towards infinity, the polygon for the base will approach the shape of the circle in which it in inscribed and the parallelogram faces will become more of a curved surface. A cylinder is a solid with congruent parallel circular bases connected by a curved surface. So, as the number of sides of the base increases infinitely, the solid becomes a cylinder.

ANSWER: a. cone b. cylinder

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41. CHALLENGE Find the volume of a cube that has a 42. WRITING IN MATH A reference sheet listed the total surface area of 54 square millimeters. formula for the surface area of a prism as SA = Bh + 2B. Use units of measure to explain why there must SOLUTION: be a typographical error in the formula. There are six congruent sides in a cube. Each side is in the shape of a square. To find the surface area of SOLUTION: the cube, find the sum of the area of each side of a Sample answer: A prism has a rectangular base 5 cube. inches long and 3 inches wide. The area of the base is B = 5 in. × 3 in. or 15 in2. If the prism is 4 inches Let a be the length of each side of a cube. So, the high, then Bh = (15 in2)(4 in.) or 60 in3. Twice the surface area of the cube is . area of the base is 2B = 2(15 in2) or 30 in2. The formula for the surface area then yields the expression SA = 60 in3 + 30 in2. The expression Bh is measured in cubic units and the expression 2B is measured in square units. Different units cannot be Take square root of each side. added, and surface area is measured in square units.

ANSWER: The length must be positive. So, Sample answer: The expression Bh is measured in The length of each edge is 3 mm. cubic units and the expression 2B is measured in square units. Different units cannot be added, and The formula for finding the volume of the prism is surface area is measured in square units. . The base of the cube is a square, so the area of the base is . The length of height is equal to the length of the side, since all the sides are congruent in a cube.

Substitute

The volume of the cube is 27 mm3.

ANSWER: 27 mm3

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43. MULTI-STEP The radius of a cylindrical vase is 5 44. GRIDDABLE An aquarium is a rectangular prism centimeters. The height of the vase is 21 centimeters. with an open top. The height and width of the Jorge fills the vase to a height of 15 centimeters, as aquarium are both 10 inches, and its length is 20 shown. inches. What is the surface area of the aquarium in square inches? a. Which of the following is the best estimate of the volume of water Jorge must add to fill the vase SOLUTION: completely? Find the surface area of a prism without the top.

A 188 cm3 B 471 cm3 C 1178 cm3 2 D 565 cm3 The total surface area of the aquarium is 800 in .

ANSWER: b. Instead of filling the vase to the brim, Jorge 800 decides to use the water in the vase to fill a cubic vase with side length 5 centimeters. How much water is left over?

c. Is the water left over enough to fill a spherical bowl of radius 6 centimeters? Explain.

SOLUTION:

a.

Choice B

b. Left over water = 1178.1 - 125 = 1053.1 cm3

c. Since the water left over from part b is over 1000, the answer is yes.

ANSWER: a. B b. 375π - 125 ≈ 1053.097 cm3 c. Yes, the radius of the bowl is 288π ≈ 904.77 whereas the water left over is 375π - 125 ≈ 1053.097 cm3

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45. ACT/SAT A stand at the state fair sells peanuts in a 46. A toy store sells beach balls with the dimensions container shaped like a square pyramid. The shown in the figure. dimensions of the container are shown.

Based on this information, which of the following statements is true? Which of the following shows the amount of peanuts F The surface area of the red beach ball is 2 times that can fit in the container? the surface area of the blue beach ball. A 260 cm3 G The surface area of the red beach ball is 4 times B 360 cm3 the surface area of the blue beach ball. H The surface area of the red beach ball is 8 times C 400 cm3 the surface area of the blue beach ball. 3 D 600 cm J The surface area of the red beach ball is 5 square E 1200 cm3 inches more than the surface area of the blue beach ball. SOLUTION: Find the volume of the container. SOLUTION: Find the surface area of each beach ball.

The volume of the container is 400 cm3. The correct choice is C.

ANSWER: C

Compare the surface areas.

Therefore, choice G is correct.

ANSWER: G

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47. Each dimension of a rectangular prism is doubled. By what factor does the surface area and volume increase, respectively?

SOLUTION: If a dimension of a prism doubles, surface area increases by 22 = 4.

If a dimension of a prism doubles, volume increases by 23 = 8.

ANSWER: 4; 8

48. The radius of a cone is doubled and the volume stays the same. What can you say about the height of the cone?

SOLUTION: If the dimension doubles, the volume of the figure increases by a factor of 23 = 8.

In order for the volume to remain the same, the height is of the original height.

ANSWER: The height is now one quarter of the original height.

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