Elementary Geometry for College Students

Daniel C. Alexander, Geralyn M. Koeberlein

Chapter 9

Surfaces and Solids - all with Video Answers

Educators

+ 2 more educators

Section 1

Prisms, Area, and Volume

Problem 1

Consider the solid shown.
a) Does it appear to be a prism?
b) Is it right or oblique?
c) What type of base(s) does the solid have?
d) Name the type of solid.
e) What type of figure is each lateral face?

James Kiss

James Kiss

Numerade Educator

Problem 2

Consider the solid shown.
a) Does it appear to be a prism?
b) Is it right or oblique?
c) What type of base(s) does the solid have?
d) Name the type of solid.
e) What type of figure is each lateral face?

James Kiss

Numerade Educator

Problem 3

Consider the hexagonal prism shown in Exercise 1.
a) How many vertices does it have?
b) How many edges (lateral edges plus base edges) does it have?
c) How many faces (lateral faces plus bases) does it have?

James Kiss

Numerade Educator

Problem 4

Consider the triangular prism shown in Exercise 2.
a) How many vertices does it have?
b) How many edges (lateral edges plus base edges) does it have?
c) How many faces (lateral faces plus bases) does it have?

James Kiss

Numerade Educator

Problem 5

If each edge of the hexagonal prism in Exercise 1 is measured in centimeters, what unit is used to measure its
(a) surface area? (b) volume?

James Kiss

Numerade Educator

Problem 6

If each edge of the triangular prism in Exercise 2 is measured in inches, what unit is used to measure its
(a) lateral area? (b) volume?

James Kiss

Numerade Educator

Problem 7

Suppose that each of the bases of the hexagonal prism in Exercise 1 has an area of $12 \mathrm{cm}^{2}$ and that each lateral face has an area of $18 \mathrm{cm}^{2} .$ Find the total (surface) area of the prism.

James Kiss

Numerade Educator

Problem 8

Suppose that each of the bases of the triangular prism in Exercise 2 has an area of 3.4 in $^{2}$ and that each lateral face has an area of 4.6 in $^{2} .$ Find the total (surface) area of the prism.

James Kiss

Numerade Educator

Problem 9

Suppose that each of the bases of the hexagonal prism in Exercise 1 has an area of $12 \mathrm{cm}^{2}$ and that the altitude of the prism measures $10 \mathrm{cm} .$ Find the volume of the prism.

James Kiss

Numerade Educator

Problem 10

Suppose that each of the bases of the triangular prism in Exercise 2 has an area of $3.4 \mathrm{cm}^{2}$ and that the altitude of the prism measures $1.2 \mathrm{cm} .$ Find the volume of the prism.

James Kiss

Numerade Educator

Problem 11

A solid is an octagonal prism.
a) How many vertices does it have?
b) How many lateral edges does it have?
c) How many base edges are there in all?

James Kiss

Numerade Educator

Problem 12

A solid is a pentagonal prism.
a) How many vertices does it have?
b) How many lateral edges does it have?
c) How many base edges are there in all?

James Kiss

Numerade Educator

Problem 13

Generalize the results found in Exercises 11 and 12 by answering each of the following questions. Assume that the number of sides in each base of the prism is $n .$ For the prism, what is the
a) number of vertices?
b) number of lateral edges?
c) number of base edges?
d) total number of edges?
e) number of lateral faces?
f) number of bases?
g) total number of faces?

James Kiss

Numerade Educator

Problem 14

In the accompanying regular pentagonal prism, suppose that each base edge measures 6 in. and that the apothem of the base measures 4.1 in. The altitude of the prism measures 10 in.
a) Find the lateral area of the prism.
b) Find the total area of the prism.
c) Find the volume of the prism.

Noah Boudrie

Numerade Educator

Problem 15

In the regular pentagonal prism shown above, suppose that each base edge measures $9.2 \mathrm{cm}$ and that the apothem of the base measures $6.3 \mathrm{cm}$. The altitude of the prism measures $14.6 \mathrm{cm}$.
a) Find the lateral area of the prism.
b) Find the total area of the prism.
c) Find the volume of the prism.

James Kiss

Numerade Educator

Problem 16

For the right triangular prism, suppose that the sides of the triangular base measure $4 \mathrm{m}, 5 \mathrm{m},$ and $6 \mathrm{m}$. The altitude is $7 \mathrm{m}$.
a) Find the lateral area of the prism.
b) Find the total area of the prism.
c) Find the volume of the prism.

Noah Boudrie

Numerade Educator

Problem 17

For the right triangular prism found in Exercise $16,$ suppose that the sides of the triangular base measure $3 \mathrm{ft}, 4 \mathrm{ft}$, and $5 \mathrm{ft}$. The altitude is $6 \mathrm{ft}$ in length.
a) Find the lateral area of the prism.
b) Find the total area of the prism.
c) Find the volume of the prism.

James Kiss

Numerade Educator

Problem 18

Given that $100 \mathrm{cm}=1 \mathrm{m},$ find the number of cubic centimeters in 1 cubic meter.

James Kiss

Numerade Educator

Problem 19

Given that 12 in. $=1 \mathrm{ft},$ find the number of cubic inches in 1 cubic foot.

James Kiss

Numerade Educator

Problem 20

Find the volume and the surface area of a "closed box" that has dimensions of 9 in., 10 in., and $1 \mathrm{ft}$.

James Kiss

Numerade Educator

Problem 21

Find the volume and the surface area of a "closed box" that has dimensions of $15 \mathrm{cm}, 20 \mathrm{cm},$ and $0.25 \mathrm{m}$ (Hint: $1 \mathrm{m}=100 \mathrm{cm} .)$

James Kiss

Numerade Educator

Problem 22

A cereal box measures 2 in. by 8 in. by 10 in. What is the volume of the box? How many square inches of cardboard make up its surface? (Disregard any hidden flaps.)

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 23

The measures of the sides of the square base of a box are twice the measure of the height of the box. If the volume of the box is 108 in $^{3},$ find the dimensions of the box.

James Kiss

Numerade Educator

Problem 24

For a given box, the height measures $4 \mathrm{m}$. If the length of the rectangular base is $2 \mathrm{m}$ greater than the width of the base and the lateral area $L$ is $96 \mathrm{m}^{2},$ find the dimensions of the box.

James Kiss

Numerade Educator

Problem 25

For the box shown, the total area is $94 \mathrm{cm}^{2} .$ Determine the value of $x$.
(FIGURE CAN'T COPY).

James Kiss

Numerade Educator

Problem 26

If the volume of the box is 252 in $^{3}$, find the value of $x .$ (See the figure for Exercise $25 .$ )

James Kiss

Numerade Educator

Problem 27

The box with dimensions indicated is to be constructed of materials that cost 1 cent per square inch for the lateral surface and 2 cents per square inch for the bases. What is the total cost of constructing the box? (FIGURE CAN'T COPY).

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 28

A hollow steel door is 32 in. wide by 80 in. tall by $1 \frac{3}{8}$ in. thick. How many cubic inches of foam insulation are needed to fill the door?

Vishal Sharma

Numerade Educator

Problem 29

A storage shed is in the shape of a pentagonal prism. The front represents one of its two pentagonal bases. What is the storage capacity (volume) of its interior? (FIGURE CAN'T COPY).

Ashley High

Numerade Educator

Problem 30

A storage shed is in the shape of a trapezoidal prism. Each trapezoid represents one of its bases. With dimensions as shown, what is the storage capacity (volume) of its interior? (FIGURE CAN'T COPY).

Breanna Ollech

Numerade Educator

Problem 31

A cube is a right square prism in which all edges have the same length. For the cube with edge $e$,
a) show that the total area is $T=6 e^{2}$
b) find the total area if $e=4 \mathrm{cm}$
c) show that the volume is $V=e^{3}$
d) find the volume if $e=4 \mathrm{cm}$
(FIGURE CAN'T COPY).

James Kiss

Numerade Educator

Problem 32

Use the formulas and drawing in Exercise 31 to find (a) the total area $T$ and (b) the volume $V$ of a cube with edges of length $5.3 \mathrm{ft}$ each.

James Kiss

Numerade Educator

Problem 33

When the length of each edge of a cube is increased by $1 \mathrm{cm},$ the volume is increased by $61 \mathrm{cm}^{3} .$ What is the length of each edge of the original cube?

James Kiss

Numerade Educator

Problem 34

The numerical value of the volume of a cube equals the numerical value of its total surface area. What is the length of each edge of the cube?

James Kiss

Numerade Educator

Problem 35

The sum of the lengths of all edges of a cube is $60 \mathrm{cm} .$ Find the volume $V$ and the surface area $T$ of the cube.

James Kiss

Numerade Educator

Problem 36

A concrete pad 4 in. thick is to have a length of $36 \mathrm{ft}$ and a width of $30 \mathrm{ft}$. How many cubic yards of concrete must be poured?

James Kiss

Numerade Educator

Problem 37

Zaidah plans a raised flower bed 2 ft high by 12 ft wide by $15 \mathrm{ft}$ long. The mulch, soil, and peat mixture used to fill the raised bed costs 15.75 dollars per cubic yard. What is the total cost of the ingredients used to fill the raised garden?

James Kiss

Numerade Educator

Problem 38

In excavating for a new house, a contractor digs a hole in the shape of a right rectangular prism. The dimensions of the hole are $54 \mathrm{ft}$ long by $36 \mathrm{ft}$ wide by $9 \mathrm{ft}$ deep. How many cubic yards of dirt were removed?

James Kiss

Numerade Educator

Problem 39

Kristine creates an open box by cutting congruent squares from the four corners of a square piece of cardboard that has a length of 24 in. per side. If the congruent squares that are removed have sides that measure 6 in. each, what is the volume of the box formed by folding and sealing the flaps? (FIGURE CAN'T COPY).

James Kiss

Numerade Educator

Problem 40

As in Exercise $39,$ find the volume of the box if four congruent squares with sides of length 6 in. are cut from the corners of a rectangular piece of poster board that is 20 in. wide by 30 in. long.

James Kiss

Numerade Educator

Problem 41

Kianna's aquarium is "box-shaped" with dimensions of 2 ft by $1 \mathrm{ft}$ by 8 in. If $1 \mathrm{ft}^{3}$ corresponds to 7.5 gal of water, what is the water capacity of her aquarium in gallons?

James Kiss

Numerade Educator

Problem 42

The gasoline tank on an automobile is "box-shaped" with dimensions of 24 in. by 20 in. by 9 in. If $1 \mathrm{ft}^{3}$ corresponds to 7.5 gal of gasoline, what is the capacity of the automobile's fuel tank in gallons?

James Kiss

Numerade Educator

Problem 43

Consider the oblique regular pentagonal prism shown. Each side of the base measures $12 \mathrm{cm},$ and the altitude measures $12 \mathrm{cm}$. (FIGURE CAN'T COPY).
Find the lateral area of the prism. (HINT: Each lateral face is a parallelogram.)

Noah Boudrie

Numerade Educator

Problem 44

Consider the oblique regular pentagonal prism shown. Each side of the base measures $12 \mathrm{cm},$ and the altitude measures $12 \mathrm{cm}$. (FIGURE CAN'T COPY).
Find the total area of the prism.

James Kiss

Numerade Educator

Problem 45

Consider the oblique regular pentagonal prism shown. Each side of the base measures $12 \mathrm{cm},$ and the altitude measures $12 \mathrm{cm}$. (FIGURE CAN'T COPY).
Find the volume of the prism.

James Kiss

Numerade Educator

Problem 46

It can be shown that the length of a diagonal of a right rectangular prism with dimensions $\ell, w,$ and $h$ is given by $d=\sqrt{\ell^{2}+w^{2}+h^{2}} .$ Use this formula to find the length of the diagonal when $\ell=12$ in., $w=4$ in., and $h=3$ in.

James Kiss

Numerade Educator

Problem 47

A diagonal of a cube joins two vertices so that the remaining points of the diagonal lie in the interior of the cube. Show the diagonal of the cube having edges of length $e$ is $e \sqrt{3}$ units long.

Jay Patel

Numerade Educator