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Geometry

Ray C. Jurgensen,Richard G. Brown,John W. Jurgensen

Chapter 12

Areas and Volumes of Solids

Educators


Problem 1

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table.

Amrita B.
Numerade Educator

Problem 2

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table.

Mitchell R.
Numerade Educator

Problem 3

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table.

Amrita B.
Numerade Educator

Problem 4

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table.

Mitchell R.
Numerade Educator

Problem 5

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table.

Amrita B.
Numerade Educator

Problem 6

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table.

Mitchell R.
Numerade Educator

Problem 7

Exercises $7-12$ refer to cubes with edges of length e. Complete the table.

Amrita B.
Numerade Educator

Problem 8

Exercises $7-12$ refer to cubes with edges of length e. Complete the table.

Mitchell R.
Numerade Educator

Problem 9

Exercises $7-12$ refer to cubes with edges of length e. Complete the table.

Amrita B.
Numerade Educator

Problem 10

Exercises $7-12$ refer to cubes with edges of length e. Complete the table.

Mitchell R.
Numerade Educator

Problem 11

Exercises $7-12$ refer to cubes with edges of length e. Complete the table.

Amrita B.
Numerade Educator

Problem 12

Exercises $7-12$ refer to cubes with edges of length e. Complete the table.

Mitchell R.
Numerade Educator

Problem 13

Find the lateral area of a right pentagonal prism with height 13 and base edges $3.2,5.8,6.9,4.7,$ and 9.4 .

Amrita B.
Numerade Educator

Problem 14

A right triangular prism has lateral area 120 $\mathrm{cm}^{2}$ . If the base edges are $4 \mathrm{cm}, 5 \mathrm{cm},$ and 6 $\mathrm{cm}$ long, find the height of the prism.

Mitchell R.
Numerade Educator

Problem 15

If the edge of a cube is doubled, the total area is multiplied by $\frac{?}{4}$ and the volume is multiplied by

Amrita B.
Numerade Educator

Problem 16

If the length, width, and height of a rectangular solid are all tripled, the lateral area is multiplied by $\underline{\text { ? }}$ , the total area is multiplied by $\underline{\text { ? }},$ and the volume is multiplied by ?

Mitchell R.
Numerade Educator

Problem 17

Facts about the base of a right prism and the height of the prism are given. Sketch each prism and find its lateral area, total area, and volume.
Equilateral triangle with side $8 ; h=10$

Amrita B.
Numerade Educator

Problem 18

Facts about the base of a right prism and the height of the prism are given. Sketch each prism and find its lateral area, total area, and volume.
Triangle with sides $9,12,15 ; h=10$

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Problem 19

Facts about the base of a right prism and the height of the prism are given. Sketch each prism and find its lateral area, total area, and volume.
Isosceles triangle with sides $13.13,10 : h=7$

Amrita B.
Numerade Educator

Problem 20

Facts about the base of a right prism and the height of the prism are given. Sketch each prism and find its lateral area, total area, and volume.
Isosceles trapezoid with bases 10 and 4 and legs $5 ; h=20$

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Problem 21

Facts about the base of a right prism and the height of the prism are given. Sketch each prism and find its lateral area, total area, and volume.
Rhombus with diagonals 6 and $8 ; h=9$

Amrita B.
Numerade Educator

Problem 22

Facts about the base of a right prism and the height of the prism are given. Sketch each prism and find its lateral area, total area, and volume.
Regular hexagon with side $8 ; h=12$

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Problem 23

The container shown has the shape of a rectangular solid. When a rock is submerged. the water level rises 0.5 $\mathrm{cm} .$ Find the volume of the rock.

Amrita B.
Numerade Educator

Problem 24

A driveway 30 $\mathrm{m}$ long and 5 $\mathrm{m}$ wide is to be paved with blacktop 3 $\mathrm{cm}$ thick. How much will the blacktop cost if it is sold at the price of $\$ 175$ per cubic meter?

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Problem 25

A brick with dimensions $20 \mathrm{cm}, 10 \mathrm{cm} .$ and 5 $\mathrm{cm}$ weighs 1.2 $\mathrm{kg} . \mathrm{A}$ second brick of the same material has dimensions $25 \mathrm{cm} .15 \mathrm{cm},$ and 4 $\mathrm{cm} .$ What is its weight?

Amrita B.
Numerade Educator

Problem 26

A drinking trough for horses is a right trapezoidal prism with dimensions shown below. If it is filled with water. about how much will the water weigh? (Hint: 1 $\mathrm{m}^{3}$ of water weighs 1 metric ton.)

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Problem 27

Find the weight. to the nearest kilogram. of the cement block shown. Cement weighs 1700 $\mathrm{kg} / \mathrm{m}^{3}$ .

Amrita B.
Numerade Educator

Problem 28

Find the weight. to the nearest 10 $\mathrm{kg} .$ of the steel I-beam shown below. Steel weighs 7860 $\mathrm{kg} / \mathrm{m}^{3}$ .

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Problem 29

Find the volume and the total surface area of each solid in terms of the given variables.

Amrita B.
Numerade Educator

Problem 30

Find the volume and the total surface area of each solid in terms of the given variables.

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Problem 31

The length of a rectangular solid is twice the width, and the height is three times the width. If the volume is $162 \mathrm{cm}^{3},$ find the total area of the solid.

Amrita B.
Numerade Educator

Problem 32

A right prism has square bases with edges that are three times as long as the lateral edges. The prism's total area is 750 $\mathrm{m}^{2}$ . Find the volume.

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Problem 33

A diagonal of a box forms a $35^{\circ}$ angle with a diagonal of the base, as shown. Use trigonometry to approximate the volume of the box.

Amrita B.
Numerade Educator

Problem 34

Refer to Exercise $33 .$ Suppose another box has a base with dimensions 8 by 6 and a diagonal that forms a $70^{\circ}$ angle with a diagonal of a base. Show that the ratio of the volumes of the two boxes is $\frac{\tan 35^{\circ}}{\tan 70^{\circ}}$ .

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Problem 35

A right prism has height $x$ and bases that are equilateral triangles with sides $x$ . Show that the volume is $\frac{1}{4} x^{3} \sqrt{3}$ .

Amrita B.
Numerade Educator

Problem 36

A right prism has height $h$ and bases that are regular hexagons with sides $s$ . Show that the volume is $\frac{3}{2} s^{2} h \sqrt{3}$ .

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Problem 37

A rectangular beam of wood 3 $\mathrm{m}$ long is cut into six pieces, as shown. Find the volume of
each piece in cubic centimeters.

Amrita B.
Numerade Educator

Problem 38

A diagonal of a cube joins two vertices not in the same face. If the diagonals are 4$\sqrt{3} \mathrm{cm}$ long, what is the volume?

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Problem 39

All nine edges of a right triangular prism are congruent. Find the length of these edges if the volume is 54$\sqrt{3} \mathrm{cm}^{3}$ .

Amrita B.
Numerade Educator

Problem 40

If the length and width of a rectangular solid are each decreased by 20$\%$ , by what percent must the height be increased for the volume to remain unchanged? Give your answer to the nearest whole percent.

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