Resources

Resources

Volume of cubes and rectangular prisms: word problems

Given the inequality:

\newline

x + 2y \geq 2

\newline

Erica's goal is to score at least

24

points in her next playground basketball game. The given inequality represents this goal, where

x

is the number of

3

-point field goals she scores and

y

is the number of

2

-point field goals she scores. If she scores

5

3

-point field goals, what is the minimum number of

2

-point field goals she must score to meet her scoring goal?

Diana is making smoothie using bananas, cantaloupes, and milk. This table gives the cost, per kilogram, of each ingredient, and the amount, in kilograms, that Diana uses: Ingredient Price per kilogram Amount Bananas

0.55

dollars per kilogram

x

kilograms Cantaloupes

y

dollar per kilogram

1.2

z

dollars per liter

p

liters The total cost of the ingredients is

2

dollars. Write an equation that relates

x

y

z

p

4

5

A regulation FIFA soccer ball is modeled after the Buckyball, a truncated icosahedron that was introduced by Buckminster Fuller. Due to air pressure, the inflated ball assumes the shape of an almost perfect sphere. A regulation soccer ball has a circumference of

\newline

70

\newline

\left(\text{cm}\right)

. Which of the following is closest to the volume of the soccer ball in cubic centimeters

\newline

(\text{cm}^3)

A company makes a greenhouse that is shaped like a rectangular prism on bottom and a half cylinder on top. A rectangular prism with a length of

3 ext{ m}

9 ext{ m}

, and a height of

2 ext{ m}

. A half cylinder is on top of the prism with the same width of

9 ext{ m}

and the same diameter as the prism's length of

3

[3 ext{ m}]

[9 ext{ m}]

[2 ext{ m}]

The rectangular prism has a base that is

[3 imes 9 ext{ m}]

and a height of

[2 ext{ m}]

. What is the volume of this greenhouse? Round to the nearest cubic meter.

\newline

81

\newline

99

\newline

117

\newline

135

Students in Mrs. Johnson's class can win prizes like colorful pencils and fun-shaped erasers for performing acts of kindness. Mrs. Johnson displays the prizes in a glass case shaped like a rectangular prism. It has a length of

18

inches, a height of

3

inches, and a volume of

270

\newline

Which equation can you use to find the width of the prize case,

w

\newline

\newline

18 \times w \times 3 = 270

\newline

270 \times 18 \times 3 = w

\newline

What is the width of the prize case?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

\newline

Students in Mrs. Johnson's class can win prizes like colorful pencils and fun-shaped erasers for performing acts of kindness. Mrs. Johnson displays the prizes in a glass case shaped like a rectangular prism. It has a length of

18

inches, a height of

3

inches, and a volume of

270

\newline

Which equation can you use to find the width of the prize case,

w

\newline

\newline

270 \times 18 \times 3 = w

\newline

18 \times w \times 3 = 270

\newline

What is the width of the prize case?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

\newline

Remove all perfect squares from inside the square root.

\newline

\sqrt{180}=

V=\frac{1}{3}\ell wh

\newline

The formula gives the volume

V

of a rectangular pyramid with base length

\ell

w

h

. What is the volume, in cubic inches, of a rectangular pyramid with a base length of

3

inches, a base width of

3

inches, and a height of

2

Ben starts walking along a path at

3\,\text{mi/h}

. One and a half hours after Ben leaves, his sister Amanda begins jogging along the same path at

6\,\text{mi/h}

. How long (in hours) will it be before Amanda catches up to Ben? Enter the exact answer. Hint: The distance formula is that distance

=

\times

time, so for example in one and a half hours, Ben has walked

3\times1.5

miles. Amanda catches up to Ben in

\square

The volume of a cone is

13,737.5 \mathrm{~m}^{3}

. If the height is

21 \mathrm{~m}

, find the radius and diameter of the cone (include correct units).

\newline

=

\newline

=

Select the information that is not needed to solve the problem. Then solve the problem.

\newline

A wallpaper border is being pasted on the walls of a rectangular room that measures

14

17 \frac{1}{2}

feet. The cost of the border is

\$ 6.50

\newline

How many feet of border is needed for the room?

\newline

Which piece of information is not needed?

\newline

a)The cost of the border.

\newline

b)The width of the room.

\newline

c)The length of the room.

\newline

The room requires

\square

feet of border.

\triangle D E F, \mathrm{~m} D E=8

\mathrm{m} E F=6

m D F=10

inches. Which lists the angles in order from smallest to largest ?

\newline

\angle F, \angle D, \angle E

\newline

\angle D, \angle F, \angle E

\newline

\angle E, \angle F, \angle D

\newline

\angle D, \angle E, \angle F

\newline

1

\newline

2

\newline

3

\newline

4

The volume of this cone is

156 \pi m^{3}

\newline

What is the height

h

\newline

Write your answer as a whole number, simplified fraction, or exact decimal.

Tessa's class had a math exam where the grades were between

0

10

\newline

N(g)

models the number of students whose grade on the exam was

g

\newline

What does the statement

\newline

N(8) > 2 \cdot N(5)

\newline

1

\newline

(A) The number of students whose grade was higher than

5

is greater than the number of students whose grade was

5

\newline

(B) The number of students whose grade was

8

is more than twice the number of students whose grade was

5

\newline

8

students whose grade was higher than twice the grade of another group of

5

Most cargo transport around the world is still done by sea. One of the largest cargo ships is the OOCL Hong Kong. It carries about

530,000

cubic feet of fuel. What is the approximate amount of fuel written in scientific notation?

\newline

\newline

5 \times 10^4

\newline

5 \times 10^5

\newline

5 \times 10^{-4}

\newline

5 \times 10^{-5}

2018

, a new record was set for the most expensive car ever sold at auction. The car, a

1962

Ferrari, sold for

48,000,000

\$

. What is the approximate price of the Ferrari, written in scientific notation?

\newline

\newline

5 \times 10^{-6}

\newline

5 \times 10^{-7}

\newline

5 \times 10^{6}

\newline

5 \times 10^{7}

A contractor is to build a new structure at a farm for seed storage. It will have a rectangular base, which is

20

35

feet long, with walls of thickness

t

feet. The structure is

90

feet tall. Excluding the volume of the walls, which of the following functions best models the volume,

V

, in cubic feet, inside the structure?

\newline

1

\newline

V(t)=63,000 t^{2}

\newline

V(t)=35 t^{2}+20 t+90

\newline

V(t)=90(35-2 t)(20-2 t)

\newline

V(t)=90(35-2 t)^{2}+20

Max wrote an algorithm that searches for a specific term within a large set of terms. The following function gives the length of the search, in number of steps, over a set with

n

\newline

S(n)=1.6 \cdot \ln (0.9 n)

\newline

What is the instantaneous rate of change of the search length for a set of

10

\newline

1

\newline

0

16

\newline

0

16

steps per second

\newline

3

5

\newline

3

5

steps per second

The following function gives the cost, in dollars, of producing

x

gallons of wood stain:

\newline

C(x)=0.0004 x^{3}-0.001 x^{2}+0.1 x+3200

\newline

What is the instantaneous rate of change of the cost when

100

gallons are produced?

\newline

1

\newline

11

9

\newline

11

9

dollars per gallon

\newline

3600

\newline

3600

dollars per gallon

Max wrote an algorithm that searches for a specific term within a large set of terms. The following function gives the length of the search, in number of steps, over a set with

n

\newline

S(n)=1.6 \cdot \ln (0.9 n)

\newline

What is the instantaneous rate of change of the search length for a set of

10

\newline

1

\newline

0

16

\newline

0

16

steps per second

\newline

3

5

\newline

3

5

steps per second

The radius of the base of a cylinder is decreasing at a rate of

12

kilometers per second.

\newline

The height of the cylinder is fixed at

2

5

\newline

At a certain instant, the radius is

40

\newline

What is the rate of change of the volume of the cylinder at that instant (in cubic kilometers per second)?

\newline

1

\newline

-1600 \pi

\newline

-2400 \pi

\newline

-4000 \pi

\newline

-360 \pi

\newline

The volume of a cylinder with radius

r

h

\pi r^{2} h

The radius of the base of a cone is decreasing at a rate of

2

centimeters per minute.

\newline

The height of the cone is fixed at

9

\newline

At a certain instant, the radius is

13

\newline

What is the rate of change of the volume of the cone at that instant (in cubic centimeters per minute)?

\newline

1

\newline

-156 \pi

\newline

-78 \pi

\newline

-507 \pi

\newline

-12 \pi

\newline

The volume of a cone with radius

r

h

\pi r^{2} \frac{h}{3}

The side of the base of a square pyramid is increasing at a rate of

6

meters per minute and the height of the pyramid is decreasing at a rate of

1

meter per minute.

\newline

At a certain instant, the base's side is

3

meters and the height is

9

\newline

What is the rate of change of the volume of the pyramid at that instant (in cubic meters per minute)?

\newline

1

\newline

-111

\newline

\mathbf{1 1 1}

\newline

-105

\newline

105

\newline

The volume of a square pyramid with base side

s

h

\frac{1}{3} s^{2} h

f=3 \pi\left(\frac{d}{2}\right)^{2} h

\newline

A board foot is a

1

1

1

inch thick volume. The formula estimates the board feet,

f

, available in a tree with a diameter of

d

inches and a height of

h

feet. Which of the following equations correctly gives the diameter in terms of the board feet and the height of the tree?

\newline

1

\newline

d=\sqrt{\frac{2 f}{3 \pi h}}

\newline

d=2 \sqrt{\frac{f}{3 \pi h}}

\newline

d=\frac{2 f^{2}}{3 \pi h}

\newline

d=\frac{f^{2} \sqrt{2}}{3 \pi h}

v=\sqrt{\frac{3 R T}{M}}

\newline

The root-mean-square speed is the measure of the speed of particles in a gas. Root-mean-square speed,

v

, can be calculated using the equation shown, where

M

is the molar mass of a gas,

R

is the molar gas constant, and

T

is the temperature. Which of the following equations correctly expresses the molar mass of a gas in terms of root-mean-square speed, temperature, and the molar gas constant?

\newline

1

\newline

M=\left(\frac{3 R T}{v}\right)^{2}

\newline

M=\frac{3 R T}{(v)^{2}}

\newline

M=\frac{(v)^{2}}{3 R T}

\newline

M=\frac{\sqrt{3 R T}}{v}

Priya is traveling by train for her job. The equation gives her distance from her destination,

x

t

hours. How long will it take for Priya to reach her destination?

\newline

1

\newline

7.7

\newline

10.1

\newline

78.0

\newline

600.6

\$ 12,000

is invested in a bond fund and the account grows to

\$ 14,309.26

4

\newline

a. Use the model

A=P e^{r t}

to determine the average rate of return under continuous compounding. Round to the nearest tenth of a percent.

\newline

b. How long will it take the investment to reach

\$ 20,000

if the rate of return continues? Round to the nearest tenth of a year.

A rectangular prism is

16

inches wide and

13

inches high. Its volume is

1,913.6

cubic inches. What is the length of the rectangular prism?

\newline

=

\square

Write the expression

16x+28

as a product using the GCF as one of the factors.

\newline

16x + 28 = \square (\square x+\square)

A box has a length of

30

inches, a width of

40

inches, and a height of

120

inches. Can a cylindrical rod with a length of

342

9

centimeters fit in the box?

\newline

\newline

The area of a piece of paper is

\frac{185}{26} \, \text{m}^2

and its breadth is

\frac{35}{13} \, \text{cm}

. Find its length and perimeter

A cylinder has a height of

3

feet and a diameter of

8

feet. What is its volume? Use

\pi \approx 3.14

and round your answer to the nearest hundredth.

\newline

\_\_\_\_\_

Mr. Wagner is smoking a turkey for Thanksgiving. The instructions for the smoker say that approximately

f(x)

hours of smoking time will be required for a turkey weighing

x

\newline

f(20) = 10

\newline

\newline

(A)It will take about

20

hours to smoke a

10

\newline

(B)It will take about

10

hours to smoke a

20

Mr. Petersen is smoking a turkey for Thanksgiving. The instructions for the smoker say that approximately

f(x)

hours of smoking time will be required for a turkey weighing

x

\newline

f(20) = 10

\newline

\newline

(A)It will take about

20

hours to smoke a

10

\newline

(B)It will take about

10

hours to smoke a

20

Mr. Brennan is smoking a turkey for Thanksgiving. The instructions for the smoker say that approximately

f(x)

hours of smoking time will be required for a turkey weighing

x

\newline

f(20) = 10

\newline

\newline

(A)It will take about

20

hours to smoke a

10

\newline

(B)It will take about

10

hours to smoke a

20

Mr. Conley is smoking a turkey for Thanksgiving. The instructions for the smoker say that approximately

f(x)

hours of smoking time will be required for a turkey weighing

x

\newline

f(20) = 10

\newline

\newline

(A)It will take about

10

hours to smoke a

20

\newline

(B)It will take about

20

hours to smoke a

10

The width of a rectangle measures

(3.7 f+3.9 g)

centimeters, and its length measures

(3.5 f+8.2 g)

centimeters. Which expression represents the perimeter, in centimeters, of the rectangle?

\newline

7.2 f+12.1

\newline

14.4 f+24.2

\newline

14.4 f+24.2 g

\newline

14.4 f+16.4 g+3.9

The width of a rectangle measures

(3.6 q-3.4)

centimeters, and its length measures

(2.1 q-6.8)

centimeters. Which expression represents the perimeter, in centimeters, of the rectangle?

\newline

5.7 q-10.2

\newline

-4.7+0.2 q

\newline

11.4 q-20.4

\newline

-9.4+0.4 q

C=40(5-s)

\newline

After Hiro's family photoshoot, the photographer sells the printed photos by the sheet. Hiro has a

\$ 200

credit from a prior session, and the equation gives the total credit left,

C

, in dollars, if he buys

s

sheets. How many sheets can Hiro purchase with his credit?

\newline

1

\newline

5

\newline

8

\newline

40

\newline

200

f(x)=\frac{x+1}{xe^{x}+e^{x}}

x\neq-1

\newline

f

is continuous for all real numbers.

\newline

f(-1)

\newline

1

\newline

1

\newline

e

\newline

2

Kudzu is a vine that was introduced to the United States from Japan in

1876

as an ornamental plant. Starting in the year

1935

, kudzu was planted throughout the Southeast to combat soil erosion. By

1946

, kudzu covered approximately

3

million acres of land in the Southeast. Sixty years later, about

7

million acres were covered by kudzu. Which of the following functions best models the amount of kudzu, in millions of acres,

t

1946

\newline

1

\newline

(A)

K(t)=3\times(1.014)^t

\newline

(B)

K(t)=3\times(0.014)^t

\newline

(C)

K(t)=3+(1.014)t

\newline

(D)

K(t)=3+(0.014)t

A cylindrical glass has a volume of approximately

502 \text{ cm}^3

. The glass has a diameter of

8 \text{ cm}

and is filled to

1\text{cm}

from the top with water. A golf ball

4\text{cm}

in diameter is placed into the glass. Complete the statements. The space remaining in the glass is [DROP DOWN

1

] and the volume of the golf ball is [DROP DOWN

2

]. Therefore, putting the golf ball in the glass of water [DROP DOWN

3

] cause the water to overflow.

24

30\%

25

milligram dose of a new antibiotic remains in the body. Which of the following functions,

M

, models the amount of the antibiotic (in milligrams) that remains in the body after

h

\newline

1

\newline

M(h)=30\times(0.3)^{\left(\frac{h}{24}\right)}

\newline

M(h)=25\times(0.3)^{\left(\frac{h}{24}\right)}

\newline

M(h)=25\times(0.3)^{h}

\newline

M(h)=25\times(0.7)^{\left(\frac{h}{24}\right)}

Creep is the ratio of elongation to original length that occurs in materials over time. For a particular steel structure, for the first

50

days the creep increases by

\newline

6\times10^{-6}

per day. After the

50

th day, when the creep is

\newline

3\times10^{-4}

, the creep triples every

62

days. After the

236

th day, how much less would the creep be if it had continued to grow linearly after the

50

\newline

1

\newline

1.1 \times10^{-3}

\newline

1.4 \times10^{-3}

\newline

6.7 \times10^{-3}

\newline

8.1 \times10^{-3}

Find the volume of a rectangular prism with a length of

5.1

\mathrm{in}

11.4

\mathrm{in}

and a height of

7.9

\mathrm{in}

, to the nearest tenth of a cubic inch.

\newline

\text{in}

^{3}

Burlington Paper Goods is considering purchasing a new delivery truck. Burlingten uses the average rate of return method to evaluate capital asset decisions.

\newline

If the initial cost of the truck is

\$67,000

with a salvage value of

\$7,000

, and it has a before tax average annual net cash flow of

\$30,000

and an annual depreciation of

\$15,000

, what is the average rate of retum on the truck? Asaume a

35\%

\newline

26.35\%

\newline

37.23\%

\newline

42.59\%

\newline

18.27\%

Burlington Paper Goods is considering purchasing a new delivery truck. Burlington uses the average rate of return method to evaluate capital asset decisions.

\newline

If the initial cost of the truck is

\$ 67,000

with a salvage value of

\$ 7,000

, and it has a before tax average annual net cash fiow of

\$ 30,000

and an annual depreciation of

\$ 15,000

, what is the average rate of retum on the truck Assume a

35 \%

\newline

26.35 \%

\newline

37.23 \%

\newline

42.59 \%

\newline

18.27 \%

Wanahton is cooking a breadstick on a rectangular baking sheet measuring

9\frac{1}{2}

13

in. Assuming the breadstick width is negligible, what is the longest breadstick Wanahton could bake by fitting it straight along the diagonal and within the baking sheet to the nearest inch?

\newline

1

\newline

13

\newline

16

\newline

124

\newline

259

A cereal company wants to enlarge the volume of the cylindrical container used for one of its products by enlarging the radius of the cylinder. The height must be

20

centimeters. The new volume of the cylinder is given by the equation

\newline

V(x)=20\pi(5+x)^{2}

\newline

x

is the additional length of the radius in centimeters. Which of the following equivalent expressions displays the current volume of the cylinder as a constant or coefficient?

\newline

1

\newline

20\pi x^{2}+200\pi x+500\pi

\newline

20\pi(x^{2}+10x+25)

\newline

20\pi(x^{2}+10x+25)

\newline

20\pi x^{2}+200\pi x+500\pi

A cellular phone salesperson earns an annual base salary of

\$25

000

. Additionally, the salesperson earns a

10\%

commission on their total sales. Which of the following functions best models the salesperson's total annual earnings,

E

, when they bill

s

dollars in total sales?

\newline

1

\newline

E(s) = 0.10 \cdot 25,000 \cdot s

\newline

E(s) = 25,000 + 0.10s

\newline

E(s) = 0.10 + 25,000s

\newline

E(s) = \frac{25,000}{0.10}s