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Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices.
SOLUTION: A polyhedron is a solid made from flat surfaces that enclose a single region of space. This solid has curved surfaces, so it is not a polyhedron. The given figure is a solid with congruent parallel circular bases connected by a curved surface. Therefore, it is a cylinder. ANSWER: not a polyhedron; cylinder
SOLUTION: The solid is formed by polygonal faces, so it is a polyhedron. It has a rectangular base and three or more triangular faces that meet at a common vertex. So, it is a rectangular pyramid. Base: Faces: Edges: Vertices, K , L , M , N , J ANSWER: a polyhedron; rectangular pyramid; base: faces edges: vertices: K, L, M, N, J Find the surface area and volume of each solid to the nearest tenth.
SOLUTION: The formulas for finding the volume and surface areas of a prism are and , where S = total surface area, V = volume, h = height of a solid, B = area of the base, P = perimeter of the base. Since the base of the prism is a rectangle, the perimeter P of the base is or 14 centimeters. The area of the base B is or 12 square centimeters. The height is 3 centimeters. The surface area of the prism is 66 square centimeters. The volume of the prism is 36 cubic centimeters. ANSWER: 66 cm^2 ; 36 cm^3
SOLUTION:
The formulas for finding the volume and surface area of a sphere are and , where S = total surface area, V = volume, and r = radius. Here, in.
The volume of the sphere is or about 904.
cubic inches.
The surface area of the sphere is or about
452.4 square inches.
ANSWER:
144π or about 452.4 in^2 ; 288π or about 904.8 in^3
5. CUPCAKES LaMea is icing cupcakes with a
cone-shaped icing bag 3.5 inches in diameter, 5
inches tall, with a slant height of approximately 5.
inches. The icing bag has no top. Find each measure
to the nearest tenth.
a. the volume of icing that will fill the bag
b. the area of plastic used to make the icing bag
SOLUTION:
a. The formula for the volume of a cone is 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 icing bag is about 16 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 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 in^2
STRUCTURE Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices.
SOLUTION: The solid is formed by polygonal faces, so it is a polyhedron. This solid has two congruent pentagonal bases, so it is a pentagonal prism. Face: Each flat surface is called face. Edges: The line 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 Edges: Vertices: ANSWER: a polyhedron; pentagonal prism; bases: ABCDE , FGHJK ; faces: ABCDE , FGHJK ; edges: vertices: A, B, C, D, E, F, G, H, J, K
SOLUTION:
A solid with all flat surfaces that enclose a single region of space is called a polyhedron. This solid has a curved surface, so it is not a polyhedron. The given figure is a solid with a circular base connected by a curved surface to a single vertex. So it is a cone. ANSWER: not a polyhedron; cone
SOLUTION: This solid is formed by polygonal faces, so it is a polyhedron. It has triangular bases. So, it is a triangular prism. Face: Each flat surface is called face. Edges: The line segments where the faces intersect are called edges. Vertex: The point where three or more edges intersect is called a vertex. Bases: Faces: Edges: Vertices: ANSWER: a polyhedron; triangular prism; bases: faces: edges: ; vertices: M, P, L, J, N, K
SOLUTION:
This solid has no faces, edges, or vertices, so it is not a polyhedron. It is a set of points in space that are the same distance from a given point. So, it is a sphere. ANSWER: not a polyhedron; sphere
SOLUTION: A solid with all flat surfaces that enclose a single region of space is called a polyhedron. The solid has a curved surface, so it is not a polyhedron. The given figure is a solid with congruent parallel circular bases connected by a curved surface. Therefore, it is a cylinder. ANSWER: not a polyhedron; cylinder
SOLUTION:
The solid is formed by polygonal faces, so it is a polyhedron. The given pyramid has a pentagonal base, so it is a pentagonal pyramid. Faces: Each flat surface is called face. Edges: The line segments where the faces intersect are called edges. Vertex: The point where three or more edges intersect is called a vertex. Base: JHGFD Faces: Edges: Vertices: ANSWER: a polyhedron; pentagonal pyramid; base: JHGFD ; faces: JHGFD , edges: vertices: J, H, G, F, D, E
SOLUTION:
The formulas for finding the volume and surface area of a cone are and , where S = total surface area, V = volume, r = radius, = slant height, and h = height. Here, the diameter of the cone is 10 yards, so the radius is 5 yards. yards and yards
The surface area of the cone is or about 282.
square yards.
The volume of the cone is about 314.2 cubic yards.
ANSWER:
90π or about 282.7 yd^2 ; 100π or about 314.2 yd^3
SOLUTION:
The formulas for finding the volume and surface area of a prism are and , where S = total surface area, V = volume, h = height, B = area of the base, and P = perimeter of the base. Since the base of the prism is a triangle, the perimeter P of the base is or 24 centimeters. The area of the base B is or 24 square centimeters. The height of the prism is 5 centimeters. The surface area of the triangular prism is 168 square centimeters. The volume of the prism is 120 cubic centimeters. ANSWER: 168 cm^2 ; 120 cm^3
SOLUTION:
The formulas for finding the volume and total surface area of a pyramid are and , where S = total surface area, V = volume, h = height, B = area of the base, P = perimeter of the base, and = slant height. 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 triangular prism is 800 square feet. The volume of the prism is 1280 cubic feet. ANSWER: 800 ft^2 ; 1280 ft^3
SOLUTION:
The formulas for finding the volume and surface area of a cylinder are and , where S = total surface area, V = volume, r = radius, and h = height. Here, mm and mm.
The surface area of the cylinder is or about
471.2 mm^2.
The volume of the cylinder is or about 785.
mm^3.
ANSWER:
150π or about 471.2 mm^2 ; 250π or about 785.4 mm^3
26. SENSE-MAKING Bento boxes are Japanese
style lunch boxes in which several different foods are
packed for lunch in varying compartments. The box
shown can be modeled by a square prism.
Assuming that the layers are filled to the top, what
volume of food can this Bento box hold?
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 volume of the lunch box is 76 cubic inches. ANSWER: 76 in^3
- ALGEBRA The surface area of a cube is 54 square inches. Find the length of each edge. SOLUTION: There are six congruent sides in a cube. Each side is in the shape 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. The length must be positive. So, The length of each edge is 3 inches. ANSWER: 3 in.
- ALGEBRA The volume of a cube is 729 cubic centimeters. Find the length of each edge. SOLUTION: 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. The length of each edge is 9 cm. ANSWER: 9 cm
- PAINTING Tara is painting her family’s fence. Each post is composed of a square prism and a square pyramid. The slant height of the pyramid is 4 inches. Determine the surface area and volume of each post. SOLUTION: Surface area of the post is equal to the surface area of the square pyramid plus the surface area of the rectangular prism. ANSWER: 1200 in^2 ; 1776 in^3
The area of the cylindrical cake to be frosted is about
113.1 in^2.
c. Divide the area to be frosted by 50.
So, 2 cans of frosting are needed for the rectangular
prism cake.
So, 3 cans of frosting are needed for the cylindrical
cake.
d. Find the surface area of the rectangular cake if
the height of the each layer 5 in.
The surface area of the rectangular cake is 152 in^2.
To find the radius of a cylindrical cake with the same height, solve the equation 152 = π r^2 + 20π r. Solving the equation using the quadratic formula gives r = –22.18 and r = 2.18. Since the radius can never be negative, r = 2.18. The same amount of frosting will be needed if the radius of the cake is 2.18 in. ANSWER: a. 96 in^2 b. 113.1 in^2 c. prism: 2 cans; cylinder: 3 cans d. 2.18 in.; if the height is 10 in., then the surface area of the rectangular cake is 152 in^2. To find the radius of a cylindrical cake with the same height, solve the equation 152 = π r^2 + 20π r. The solutions are r = –22.18 or r = 2.18. Using a radius of 2.18 in. gives surface area of about 152 in^2.
- CHANGING UNITS A gift box has a surface area of 6.25 square feet. What is the surface area of the box in square inches? SOLUTION: Surface area of the gift box = 6.25 ft^2 1 foot = 12 inches Surface area of the gift box = 6.25(12 inches)^2 in^2 = 900 in^2 ANSWER: 900 in^2
- CHANGING UNITS A square pyramid has a volume of 4320 cubic inches. What is the volume of this pyramid in cubic feet? SOLUTION: Volume of the pyramid = 4320 in^3 1 foot = 12 inches So, 1 inch = foot. Volume of a pyramid = ft^3 = 2.5 ft^3 ANSWER: 2.5 ft^3
- EULER’S FORMULA The number of faces F , vertices V , and edges E of a polyhedron are related by Euler’s (OY luhrz) Formula: F + V = E + 2. Determine whether Euler’s Formula is true for each of the figures in Exercises 18–23. SOLUTION: Use Euler’s formula: F + V = E + 2 Exercise 18: Rectangular Prism: 6 + 8 = 12 + 2 So, Euler’s formula is true. The answer is “Yes”. Exercise 19: 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 Formula does not apply. So, the answer is “No”. Exercise 21: Triangular Prism: 5 + 6 = 9 + 2 So, Euler’s formula is true. The answer is “Yes”. Exercise 22: Square Pyramid: 5 +5 = 8 + So, Euler’s formula is true. The answer is “Yes”. Exercise 23: This figure is a cylinder and not a polyhedron, so Euler’s Formula does not apply. So, the answer is “No”. ANSWER: Exercise 18: yes, 6 + 8 = 12 + 2; Exercise 19: yes, 6
- 8 = 12 + 2;Exercise 20: no, this figure is not a polyhedron, so Euler’s Formula does not apply; Exercise 21: yes, 5 + 6 = 9 + 2; Exercise 22: yes, 5 + 5 = 8 + 2; Exercise 23: no, this figure is not a polyhedron, so Euler’s Formula does not apply.
- CHANGING DIMENSIONS A rectangular prism has a length of 12 centimeters, width of 18 centimeters, and height of 22 centimeters. Describe the effect on the volume of a rectangular prism when each dimension is doubled. SOLUTION: The formula for finding the volume of a prism is , where V = volume, h = height, and B = area of the base. Since the base of the prism is a rectangle, the area of the base B is or 216 square centimeters. Here, height of the prism = 22 cm. The volume of the original prism is 4752 cm^3. Double the dimensions and find the volume. Volume of the new prism The volume increased by a factor of 8 when each dimension was doubled. ANSWER: The volume of the original prism is 4752 cm^3. The volume of the new prism is 38,016 cm^3. The volume increased by a factor of 8 when each dimension was doubled.
- MULTIPLE REPRESENTATIONS In this 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
Doubling the radius results in an increase in the volume by a factor of 4. d.
- CRITIQUE Alex and Emily are calculating the surface area of the rectangular prism shown. Is either of them correct? Explain your reasoning. 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. Here, height of the prism = 3 ft. The total surface area of the prism is 94 in^2. So, both answers are incorrect. ANSWER: Neither; sample answer: the surface area is twice the sum of the areas of the top, front, and left side of the prism or 2(5 · 3 + 5 · 4 + 3 · 4), which is 94 in^2.
- REASONING Is a cube a regular polyhedron? Explain. SOLUTION: In a cube, all of the faces are regular congruent squares 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.
- CHALLENGE Describe the solid that results if the number of sides of each base increases infinitely. The bases of each solid are regular polygons inscribed in a circle. a. pyramid b. prism SOLUTION: 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 polygon 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 circular base connected by a curved surface to a 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 parallelogram 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
- OPEN-ENDED Draw an irregular 14-sided polyhedron which has two congruent bases. SOLUTION: Sample answer: If the bases are congruent are congruent, there must be 12 faces connecting the bases to make a total of 14. Therefore, the two bases must be congruent 12-sided polygons. ANSWER: Sample answer:
- MULTI-STEP The radius of a cylindrical vase is 5 centimeters. The height of the vase is 21 centimeters. Jorge fills the vase to a height of 15 centimeters, as shown. a. Which of the following is the best estimate of the volume of water Jorge must add to fill the vase completely? A 188 cm^3 B 471 cm^3 C 1178 cm^3 D 565 cm^3 b. Instead of filling the vase to the brim, Jorge 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 cm^3 c. Since the water left over from part b is over 1000, the answer is yes. ANSWER: a. B b. 375π - 125 ≈ 1053.097 cm^3 c. Yes, the radius of the bowl is 288π ≈ 904. whereas the water left over is 375π - 125 ≈ 1053. cm^3
- GRIDDABLE An aquarium is a rectangular prism with an open top. The height and width of the aquarium are both 10 inches, and its length is 20 inches. What is the surface area of the aquarium in square inches? SOLUTION: Find the surface area of a prism without the top. The total surface area of the aquarium is 800 in^2. ANSWER: 800
- ACT/SAT A stand at the state fair sells peanuts in a container shaped like a square pyramid. The dimensions of the container are shown. Which of the following shows the amount of peanuts that can fit in the container? A 260 cm^3 B 360 cm^3 C 400 cm^3 D 600 cm^3 E 1200 cm^3 SOLUTION: Find the volume of the container. The volume of the container is 400 cm^3. The correct choice is C. ANSWER: C
- A toy store sells beach balls with the dimensions shown in the figure. Based on this information, which of the following statements is true? F The surface area of the red beach ball is 2 times the surface area of the blue beach ball. G The surface area of the red beach ball is 4 times the surface area of the blue beach ball. H The surface area of the red beach ball is 8 times the surface area of the blue beach ball. J The surface area of the red beach ball is 5 square inches more than the surface area of the blue beach ball. SOLUTION: Find the surface area of each beach ball. Compare the surface areas. Therefore, choice G is correct. ANSWER: G
