Applied Geometry Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2

Question 21Question →

An interior designer is planning to cover a rectangular accent wall with custom wallpaper. The wall measures $15\text{ feet}$ in width and $8\text{ feet}$ in height. If the wallpaper costs $\$4$ per square foot, what is the total cost, in dollars, of the wallpaper needed for this wall?

Show answer & explanation

Answer: 480

Answer

480

The wall is a rectangle, so its area is found by multiplying the width (

15\text{ feet}

) by the height (

8\text{ feet}

), which equals

120\text{ square feet}

. Since the wallpaper costs

\$4

per square foot, multiplying the total area by the rate (

120 \times 4

) gives a total cost of

\$480

.

Step-by-Step Solution

1

Calculate the area of the rectangular wall.

15 \times 8 = 120\text{ square feet}

The area determines how much surface the wallpaper needs to cover.

2

Calculate the total cost of the wallpaper.

120 \times 4 = 480\text{ dollars}

Multiplying the total square footage by the cost per square foot provides the final amount.

Key Concept

Calculating rectangular area and applying a unit cost rate.

Question 22Question →

A rectangular reservoir for a water purification system measures $20$ feet long, $15$ feet wide, and $10$ feet deep. Safety protocols dictate that the reservoir must never exceed $90\%$ of its total volume capacity. The reservoir currently contains $900$ cubic feet of water. If a high-capacity pump adds water to the reservoir at a rate of $60$ cubic feet per minute, how many minutes will it take for the water to reach the maximum allowed capacity?

Show answer & explanation

Answer: 30

Answer

30

First, the total volume of the rectangular reservoir is found by multiplying its length, width, and depth (

20 \times 15 \times 10 = 3000

cubic feet). Next, the maximum allowed capacity is determined by finding

90\%

of the total volume (

0.90 \times 3000 = 2700

cubic feet). Since the reservoir already contains

900

cubic feet of water, the exact amount needed to reach the maximum safe capacity is the difference (

2700 - 900 = 1800

cubic feet). Finally, the time required to add this volume at a rate of

60

cubic feet per minute is calculated by dividing the needed volume by the pump's rate (

1800 \div 60 = 30

minutes).

Step-by-Step Solution

1

Calculate the total volume of the reservoir.

20 \times 15 \times 10 = 3000

cubic feet

The total volume is required to determine what the absolute maximum capacity is before applying the percentage limit.

2

Calculate the maximum allowed capacity.

3000 \times 0.90 = 2700

cubic feet

Safety protocols dictate the reservoir cannot exceed

90\%

of its total physical volume.

3

Determine the required volume of water to add.

2700 - 900 = 1800

cubic feet

The reservoir already contains

900

cubic feet of water, so we only need to pump the difference to reach the safe limit.

4

Calculate the time required to fill the reservoir to the required level.

1800 \div 60 = 30

minutes

Time is calculated by dividing the total volume to be added by the filling rate of the pump.

Key Concept

Calculating rectangular volume, applying percentage constraints to capacity, and solving for time using a given rate.

Question 23Question →

A homeowner is planning to paint a circular concrete patio. The patio has a diameter of $6$ meters. What is the total area of the patio that needs to be painted? (Use $\pi = 3.14$ )

9.42

square meters

18.84

square meters

28.26

square meters

113.04

square meters

Show answer & explanation

Answer: $28.26$ square meters

Answer

The correct area that needs to be painted is

28.26

square meters.

The correct answer is found by first determining the radius, which is half of the diameter (

6 \div 2 = 3

meters). Next, substitute the radius into the area formula

A = \pi \times r^2

. Squaring the radius gives

3^2 = 9

. Multiplying by

\pi

gives

3.14 \times 9 = 28.26

square meters.

Step-by-Step Solution

1

Find the radius of the circular patio from the given diameter.

r = 6 \div 2 = 3

meters

The area formula requires the radius, which is exactly half of the diameter.

2

Identify the formula for the area of a circle.

A = \pi \times r^2

The problem asks for the total surface area to be painted.

3

Substitute the radius and the given value of

\pi

into the area formula and solve.

A = 3.14 \times (3)^2 = 3.14 \times 9 = 28.26

square meters

This calculation provides the final area in square meters.

Key Concept

Calculating the area of a circle given its diameter.

Question 24Question →

An engineering unit is excavating a rectangular trench that is $12$ yards long, $6$ yards wide, and $9$ feet deep. The excavated dirt will be removed using dump trucks. If each dump truck has a rectangular bed that measures $12$ feet long, $6$ feet wide, and $3$ feet deep, what is the total number of truckloads required to remove all the dirt from the trench?

81

9

27

243

Show answer & explanation

Answer: 27

Answer

27

To find the number of truckloads, the total volume of the trench must be divided by the volume of one truck, ensuring all measurements use the same unit. Converting all dimensions to yards, the truck volume is

4 \times 2 \times 1 = 8

cubic yards, and the trench volume is

12 \times 6 \times 3 = 216

cubic yards. Dividing

216

by

8

yields exactly

27

truckloads.

Step-by-Step Solution

1

Convert the dimensions of the dump truck bed from feet to yards.

Length =

4

yards, Width =

2

yards, Depth =

1

yard.

Standardizing all measurements to a single unit (yards) is necessary before volumes can be accurately compared.

2

Calculate the volume capacity of a single dump truck in cubic yards.

4 \times 2 \times 1 = 8

cubic yards.

Finding the volume of the rectangular bed determines exactly how much dirt one truck can haul per trip.

3

Convert the depth of the trench from feet to yards.

9

feet

\div 3 = 3

yards.

The length and width are given in yards, so the depth must also be in yards to calculate the correct total volume.

4

Calculate the total volume of the rectangular trench in cubic yards.

12 \times 6 \times 3 = 216

cubic yards.

Multiplying length by width by depth gives the total amount of dirt that needs to be removed.

5

Determine the total number of truckloads required.

216 \div 8 = 27

truckloads.

Dividing the total trench volume by the capacity of a single truck yields the required number of trips.

Key Concept

Calculating and comparing volumes of rectangular prisms while performing correct unit conversions between standard linear measurements.

Question 25Question →

An event coordinator is installing a temporary dance floor in a rectangular banquet room measuring $60\text{ feet}$ long and $45\text{ feet}$ wide. A fixed rectangular dining section that measures $30\text{ feet}$ by $15\text{ feet}$ will remain uncovered. If the temporary flooring material costs $\$4$ per square yard, what is the total cost of the flooring?

\$480

\$1,000

\$3,000

\$9,000

Show answer & explanation

Answer: $\$1,000$

Answer

The total cost of the flooring is

\$1,000

.

The correct answer is found by first calculating the net area to be covered in square feet (

2,700 - 450 = 2,250\text{ sq ft}

). Then, this area must be converted to square yards by dividing by

9

(since

1\text{ square yard} = 9\text{ square feet}

), resulting in

250\text{ square yards}

. Finally, multiplying by the rate of

\$4

per square yard yields

\$1,000

.

Step-by-Step Solution

1

Calculate the total area of the banquet room.

60\text{ feet} \times 45\text{ feet} = 2,700\text{ square feet}

To find the maximum possible floor space before accounting for the uncovered section.

2

Calculate the area of the fixed dining section.

30\text{ feet} \times 15\text{ feet} = 450\text{ square feet}

To determine how much space will not receive the temporary flooring.

3

Subtract the uncovered area from the total room area.

2,700 - 450 = 2,250\text{ square feet}

To find the net area that requires the temporary dance floor.

4

Convert the net area from square feet to square yards.

2,250 \div 9 = 250\text{ square yards}

Because

1\text{ square yard} = 9\text{ square feet}

(

3\text{ ft} \times 3\text{ ft}

), and the material is priced per square yard.

5

Calculate the total cost.

250\text{ sq yd} \times \$4/\text{sq yd} = \$1,000

To find the final expense by multiplying the total required square yards by the price per square yard.

Key Concept

Calculating net rectangular area and performing two-dimensional unit conversions.

Question 26Question →

A cylindrical well is being drilled into the ground. The well has a diameter of $36\text{ inches}$ and a target depth of $70\text{ feet}$ . If soil is extracted from the well at a constant rate of $11\text{ cubic feet per hour}$ , how many hours will it take to drill the well completely? (Use $\frac{22}{7}$ for $\pi$ )

Show answer & explanation

Answer: 45

Answer

45

The correct answer is found by first ensuring all measurements are in feet. A diameter of

36\text{ inches}

is equal to

3\text{ feet}

, making the radius

1.5\text{ feet}

(or

\frac{3}{2}\text{ feet}

). Using the formula

V = \pi r^2 h

, the total volume of soil is

\frac{22}{7} \times \left(\frac{3}{2}\right)^2 \times 70 = 495\text{ cubic feet}

. Finally, dividing the total volume by the extraction rate of

11\text{ cubic feet per hour}

gives exactly

45\text{ hours}

.

Step-by-Step Solution

1

Convert the diameter to feet

36\text{ inches} = 3\text{ feet}

Units must be consistent (feet) with the depth and the extraction rate before calculating volume.

2

Determine the radius of the well

r = 1.5\text{ feet}

(or

\frac{3}{2}\text{ feet}

)

The volume formula requires the radius, which is half of the diameter.

3

Calculate the volume of the cylindrical well

V = \frac{22}{7} \times \left(\frac{3}{2}\right)^2 \times 70 = \frac{22}{7} \times \frac{9}{4} \times 70 = 495\text{ cubic feet}

This determines the total amount of soil that needs to be extracted.

4

Calculate the time required to drill the well

\text{Time} = \frac{495}{11} = 45\text{ hours}

Time is equal to the total volume divided by the rate of extraction.

Key Concept

Calculating the volume of a cylinder with unit conversions and applying it to a rate problem.

Question 27Question →

A circular water pipe has a diameter of $8$ inches. What is the cross-sectional area of the pipe? Use $\pi = 3.14$ .

Show answer & explanation

Answer: 50.24

Answer

50.24 square inches

The area of a circle is calculated using the formula

A = \pi r^2

. First, find the radius by dividing the diameter by 2 (

8 \div 2 = 4

). Then, substitute

r = 4

and

\pi = 3.14

into the formula:

A = 3.14 \times 4^2 = 3.14 \times 16 = 50.24

.

Step-by-Step Solution

1

Find the radius of the pipe.

Radius = 4 inches

The formula for the area of a circle requires the radius, which is exactly half of the diameter.

2

Calculate the cross-sectional area using the area formula.

Area = 50.24 square inches

Squaring the radius (16) and multiplying by 3.14 gives the total area of the circle.

Key Concept

Calculating the area of a circle given its diameter.

Question 28Question →

A contractor is applying a specialized primer to a rectangular warehouse wall that is $60\text{ feet}$ long and $18\text{ feet}$ high. The wall contains a rectangular loading bay door measuring $30\text{ feet}$ long and $12\text{ feet}$ high, which will not receive the primer. If the primer material costs $\$15$ per square yard, what is the total cost, in dollars, of the primer needed for the wall?

Show answer & explanation

Answer: 1200

Answer

1200

The total gross area of the wall is

60 \times 18 = 1080\text{ square feet}

. The area of the unpainted loading door is

30 \times 12 = 360\text{ square feet}

. Subtracting the door's area yields a net area of

720\text{ square feet}

to be primed. Because

1\text{ square yard}

equals

9\text{ square feet}

, the net area in square yards is

720 \div 9 = 80\text{ square yards}

. Finally, multiplying the

80\text{ square yards}

by the

\$15

cost per square yard gives a total cost of

\$1200

.

Step-by-Step Solution

1

Calculate the gross area of the warehouse wall.

1080\text{ sq ft}

Multiply the length (

60\text{ feet}

) by the height (

18\text{ feet}

) to find the total rectangular area before accounting for the door.

2

Calculate the area of the loading bay door.

360\text{ sq ft}

Multiply the door's length (

30\text{ feet}

) by its height (

12\text{ feet}

) to determine the area that will not be painted.

3

Find the net area to be primed.

720\text{ sq ft}

Subtract the area of the door from the gross wall area (

1080 - 360

) to find the exact surface area requiring primer.

4

Convert the net square footage into square yards.

80\text{ sq yd}

Since the cost is given per square yard, and there are

9\text{ square feet}

in

1\text{ square yard}

(

3\text{ ft} \times 3\text{ ft}

), divide the net area by 9 (

720 \div 9

).

5

Calculate the total cost of the primer.

\$1200

Multiply the required area in square yards (

80

) by the cost per square yard (

\$15

).

Key Concept

Calculating net rectangular area via subtractive cutouts and applying square unit conversions.

Question 29Question →

A circular helicopter landing pad has a radius of $20$ feet. A reflective border needs to be painted exactly along its outer edge. What is the total length of the painted border? (Use $\pi = 3.14$ )

125.6

feet

62.8

feet

1,256

feet

12.56

feet

Show answer & explanation

Answer: $125.6$ feet

Answer

125.6

feet

The total length of the border around the outside of a circle is its circumference. Because the radius is provided as

20

feet, we use the formula

C = 2\pi r

. Substituting the given values, we get

2 \times 3.14 \times 20 = 125.6

feet.

Step-by-Step Solution

1

Identify the correct geometric property to calculate.

Circumference is needed.

The painted border goes along the 'outer edge' of the circular pad, which corresponds to the circumference.

2

Select the appropriate formula for circumference.

C = 2\pi r

The radius (

r

) is given in the problem.

3

Substitute the given values and calculate.

C = 2 \times 3.14 \times 20 = 125.6

feet.

This yields the total length of the border needed.

Key Concept

Calculating the circumference of a circle given its radius for a real-world application.

Estimated Time:45s

Question 30Question →

A hotel director is purchasing custom carpet for a rectangular conference room that measures $45\text{ feet}$ in length and $36\text{ feet}$ in width. The carpet is priced at $\$15$ per square yard. What is the total cost of the carpet needed to completely cover the room?

\$810

\$2,700

\$8,100

\$24,300

Show answer & explanation

Answer: $\$2,700$

Answer

\$2,700

First, the area of the rectangular room is found by multiplying the length and width:

45\text{ feet} \times 36\text{ feet} = 1,620\text{ square feet}

. Next, because the carpet is priced per square yard, the area must be converted. Since

1\text{ square yard}

equals

9\text{ square feet}

, divide the area by 9:

1,620 \div 9 = 180\text{ square yards}

. Finally, multiply the

180\text{ square yards}

by the unit price of

\$15

to get the total cost of

\$2,700

.

Step-by-Step Solution

1

Calculate the total area of the room in square feet.

45\text{ feet} \times 36\text{ feet} = 1,620\text{ square feet}

The area formula for a rectangle (length multiplied by width) determines the total floor space to be covered.

2

Convert the area from square feet to square yards.

1,620 \div 9 = 180\text{ square yards}

The carpet is priced per square yard. Because

1\text{ yard} = 3\text{ feet}

,

1\text{ square yard} = 3\text{ feet} \times 3\text{ feet} = 9\text{ square feet}

. You must divide by 9.

3

Calculate the total cost by multiplying the area in square yards by the price per square yard.

180\text{ square yards} \times \$15\text{/square yard} = \$2,700

Multiplying the total required material by the unit price gives the final cost.

Key Concept

Calculating rectangular area and converting between two-dimensional units (square feet to square yards) before applying rates.

Question 31Question →

A landscaper is designing a circular flower bed with a diameter of $10$ feet. She needs to install a flexible plastic border completely around the outer edge of the flower bed. What is the total length, in feet, of the plastic border needed? Use $\pi = 3.14$ .

Show answer & explanation

Answer: 31.4

Answer

31.4

The problem asks for the length of a border around a circle, which is the circumference. The circumference is found by multiplying the diameter (

10

feet) by

\pi

(

3.14

), resulting in

31.4

feet.

Step-by-Step Solution

1

Determine the required measurement

Calculate the circumference

The plastic border goes around the outer edge of the flower bed.

2

Select the correct formula

C = \pi \times d

We are given the diameter (

10

feet) and need the circumference.

3

Calculate the circumference

3.14 \times 10 = 31.4

Substitute

\pi = 3.14

and

d = 10

into the formula.

Key Concept

Circumference of a circle

Estimated Time:45s

Question 32Question →

A rescue helicopter flies 12 miles due south and then 35 miles due west to reach a medical emergency. After the pickup, the helicopter flies directly back to its base in a straight line. What is the total distance the helicopter traveled during this entire trip?

37 miles

47 miles

84 miles

94 miles

Show answer & explanation

Answer: 84 miles

Answer

84 miles

The total distance is the sum of the outbound journey and the return journey. The outbound journey consists of the two legs of the flight: 12 + 35 = 47 miles. The return journey is a straight line that forms the hypotenuse of a right triangle. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), the return distance is

\sqrt{12^2 + 35^2} = \sqrt{1369} = 37

miles. The total distance traveled is 47 + 37 = 84 miles.

Step-by-Step Solution

1

Calculate the distance traveled during the outbound trip.

12 + 35 = 47

miles

The helicopter travels along two perpendicular paths (south and west) to reach the destination.

2

Set up the Pythagorean theorem to find the straight-line return distance.

c^2 = 12^2 + 35^2

The south and west paths form a right angle, making the direct return path the hypotenuse of a right triangle.

3

Solve for the return distance (the hypotenuse).

c = \sqrt{144 + 1225} = \sqrt{1369} = 37

miles

Taking the square root of the sum of the squares gives the length of the straight-line return trip.

4

Add the outbound distance and the return distance.

47 + 37 = 84

miles

The question asks for the total distance traveled during the entire round trip.

Key Concept

Applying the Pythagorean theorem to find a missing distance in a multi-step navigation problem.

Question 33Question →

A logistics company is packing a new rectangular shipping crate. The crate measures $10$ feet long, $4$ feet wide, and $2$ feet high. What is the total volume of the crate in cubic feet?

28

40

80

800

Show answer & explanation

Answer: 80

Answer

80

The total volume of a rectangular crate is found by multiplying its length, width, and height. Multiplying

10 \times 4 \times 2

gives

80

cubic feet.

Step-by-Step Solution

1

Identify the given dimensions of the rectangular crate.

Length = 10, Width = 4, Height = 2

These are the values needed to use the volume formula.

2

Apply the formula for the volume of a rectangular prism: Volume = length × width × height.

10 × 4 × 2

This formula calculates the total 3D space inside the rectangular container.

3

Multiply the numbers together to find the final volume.

10 × 4 = 40, and 40 × 2 = 80

Completing the multiplication provides the volume in cubic feet.

Key Concept

The volume of a rectangular container is calculated by multiplying its length, width, and height.

Question 34Question →

A university is installing artificial turf on an indoor practice facility. The rectangular facility measures $120\text{ feet}$ in length and $60\text{ feet}$ in width. Inside the facility, there are two identical rectangular equipment zones, each measuring $30\text{ feet}$ long and $15\text{ feet}$ wide, which will remain bare concrete and will not be covered with turf. If the artificial turf costs $\$24$ per square yard, what is the total cost, in dollars, of the artificial turf needed for the facility?

Show answer & explanation

Answer: 16800

Answer

16800

The correct answer is found by calculating the total area of the facility (

7,200\text{ sq ft}

), subtracting the area of the two equipment zones (

900\text{ sq ft}

), converting the remaining area (

6,300\text{ sq ft}

) from square feet to square yards by dividing by 9 (

700\text{ sq yd}

), and finally multiplying by the cost per square yard (

24) to get

16,800.

Step-by-Step Solution

1

Calculate the total area of the facility.

120 \times 60 = 7,200\text{ sq ft}

To establish the maximum possible area before exclusions.

2

Calculate the area of the two equipment zones that will not be turfed.

2 \times (30 \times 15) = 900\text{ sq ft}

The two zones are bare concrete, so their combined area must be subtracted from the total.

3

Subtract the equipment zones from the total facility area.

7,200 - 900 = 6,300\text{ sq ft}

To find the actual area that requires artificial turf.

4

Convert the turfed area from square feet to square yards.

6,300 \div 9 = 700\text{ sq yd}

There are 9 square feet in one square yard (

3\text{ ft} \times 3\text{ ft}

), and the turf is priced per square yard.

5

Calculate the total cost.

700 \times 24 = \$16,800

Multiplying the required square yards by the price per square yard gives the final cost.

Key Concept

Calculating area with subtractive regions and converting between square units.

Question 35Question →

A community center is building a new rectangular sandbox for the playground. The sandbox measures $6$ feet long, $4$ feet wide, and $1.5$ feet deep. How many cubic feet of sand are required to fill the sandbox completely to the top?

Show answer & explanation

Answer: 36

Answer

36

The correct answer is found by multiplying the three dimensions of the rectangular sandbox (

6

feet,

4

feet, and

1.5

feet) together, which yields exactly

36

cubic feet of total capacity.

Step-by-Step Solution

1

Identify the correct volume formula for a rectangular container.

V = l \times w \times h

The total capacity of a rectangular sandbox is calculated by multiplying its three spatial dimensions together.

2

Substitute the provided measurements into the formula.

V = 6 \times 4 \times 1.5

The given dimensions are a length of 6 feet, a width of 4 feet, and a depth of 1.5 feet.

3

Multiply the dimensions to find the total volume.

24 \times 1.5 = 36

Calculating the final cubic footage required to completely fill the sandbox.

Key Concept

Calculating the volume of a rectangular container using the formula

V = l \times w \times h

.

Question 36Question →

An architect's blueprint shows a circular courtyard with a diameter of $2$ inches. The scale on the blueprint is $1 \text{ inch} = 6 \text{ yards}$ . What is the actual area of the courtyard in square yards? (Use $\pi = 3.14$ )

11.304

18.84

113.04

452.16

Show answer & explanation

Answer: $113.04$

Answer

The actual area of the courtyard is

113.04

square yards.

First, the blueprint diameter of

2

inches must be converted to the actual diameter using the scale

1 \text{ inch} = 6 \text{ yards}

. This gives an actual diameter of

12

yards. Next, the radius is found by halving the diameter, resulting in

6

yards. Finally, applying the area formula

A = \pi \times r^2

, the calculation is

3.14 \times 6^2 = 3.14 \times 36 = 113.04

square yards.

Step-by-Step Solution

1

Use the scale to find the actual diameter of the courtyard.

2 \text{ inches} \times 6 \text{ yards/inch} = 12 \text{ yards}

Before calculating the actual area, the blueprint dimensions must be converted to actual dimensions.

2

Find the actual radius.

12 \text{ yards} \div 2 = 6 \text{ yards}

The area formula for a circle requires the radius, which is half of the diameter.

3

Calculate the area using the formula

A = \pi \times r^2

.

A = 3.14 \times (6)^2 = 3.14 \times 36 = 113.04 \text{ square yards}

This determines the total two-dimensional space the courtyard will occupy.

Key Concept

Calculating circular area from a scale drawing requires converting the linear dimensions first, and ensuring the radius is used instead of the diameter in the formula

A = \pi r^2

.

Question 37Question →

A landscaper is preparing a new rectangular garden planter bed. The planter bed is $8$ feet long, $3$ feet wide, and $24$ inches deep. How many cubic feet of soil are needed to completely fill the planter bed?

24

44

48

576

Show answer & explanation

Answer: $48$

Answer

The correct volume is

48

cubic feet.

The correct answer accurately converts the depth to

2

feet before applying the volume formula. Multiplying the uniform dimensions (

8 \text{ ft} \times 3 \text{ ft} \times 2 \text{ ft}

) yields a total of

48

cubic feet.

Step-by-Step Solution

1

Convert the depth from inches to feet so all units match.

24 \text{ inches} \div 12 \text{ inches/foot} = 2 \text{ feet}

.

Volume calculations require all dimensions to be in the same unit of measurement (feet, since the answer asks for cubic feet).

2

Calculate the volume of the rectangular planter bed using the formula

V = L \times W \times H

.

8 \text{ feet} \times 3 \text{ feet} \times 2 \text{ feet} = 48 \text{ cubic feet}

.

Multiplying the length, width, and depth together gives the total 3D capacity of a rectangular container.

Key Concept

To find the volume of a rectangular prism, ensure all measurements are in the same unit, then multiply the length, width, and height.

Question 38Question →

A supply sergeant is reviewing the specifications for a rectangular storage pod. The pod has a known total volume of $120$ cubic feet. If the pod measures $6$ feet in length and $4$ feet in width, what is the height of the pod in feet?

Show answer & explanation

Answer: 5

Answer

5

The volume of a rectangular shape is determined by multiplying its length, width, and height (

V = L \times W \times H

). Given a total volume of

120

cubic feet, a length of

6

feet, and a width of

4

feet, the equation becomes

120 = 6 \times 4 \times H

. First, find the area of the base by multiplying the length and width (

6 \times 4 = 24

). Finally, divide the total volume by this base area (

120 \div 24

) to find the missing height, which is

5

feet.

Step-by-Step Solution

1

Identify the formula for the volume of a rectangular container.

V = L \times W \times H

This standard formula relates the total volume to the object's three dimensions.

2

Substitute the given values into the formula.

120 = 6 \times 4 \times H

We are given the total volume (

120

), the length (

6

), and the width (

4

).

3

Calculate the area of the base by multiplying the length and the width.

6 \times 4 = 24

Simplifying the right side of the equation reduces it to a single multiplication step.

4

Solve for the height by dividing the total volume by the base area.

120 \div 24 = 5

Isolating

H

provides the missing dimension.

Key Concept

Solving for a missing dimension of a rectangular prism using the volume formula.

Estimated Time:45s

Question 39Question →

A military transport vehicle has circular tires with a diameter of $4$ feet. If the vehicle moves forward such that each tire makes exactly $250$ full revolutions without slipping, what is the total distance traveled in feet? Use $\pi = 3.14$ .

Show answer & explanation

Answer: 3140

Answer

3140

Each time a tire completes one full revolution, the vehicle moves forward a distance exactly equal to the tire's circumference. The circumference is found by multiplying the diameter (

4

feet) by

\pi

(

3.14

), yielding

12.56

feet per revolution. Multiplying this linear distance by

250

revolutions gives a total distance of

3140

feet.

Step-by-Step Solution

1

Calculate the circumference of the tire.

12.56 feet

One full revolution of a tire covers a linear distance exactly equal to its circumference.

2

Calculate the total distance traveled by multiplying the circumference by the number of revolutions.

3140 feet

The total distance is the distance covered in one revolution multiplied by the total number of revolutions.

Key Concept

Circumference and Wheel Revolutions

Question 40Question →

A cylindrical oil storage tank has a diameter of $20\text{ feet}$ and a height of $5\text{ feet}$ . Using $3.14$ for $\pi$ , what is the total volume of the tank?

157\text{ cubic feet}

314\text{ cubic feet}

1,570\text{ cubic feet}

6,280\text{ cubic feet}

Show answer & explanation

Answer: $1,570\text{ cubic feet}$

Answer

The total volume of the tank is

1,570\text{ cubic feet}

.

The correct answer is found by first halving the given diameter (20 feet) to get a radius of 10 feet. The radius is then squared (100) and multiplied by the height (5) and pi (3.14) according to the volume formula

V = \pi r^2 h

. The calculation

3.14 \times 100 \times 5

equals

1,570\text{ cubic feet}

.

Step-by-Step Solution

1

Find the radius of the cylindrical tank.

r = \frac{d}{2} = \frac{20}{2} = 10\text{ feet}

The volume formula for a cylinder requires the radius, not the diameter.

2

Set up the volume formula for a cylinder.

V = \pi r^2 h

This is the standard geometric formula to find the volume of any cylinder.

3

Substitute the known values into the formula and solve.

V = 3.14 \times (10)^2 \times 5 = 3.14 \times 100 \times 5 = 314 \times 5 = 1,570\text{ cubic feet}

Squaring 10 gives 100, which moves the decimal point in 3.14 two places to the right, making the final multiplication straightforward.

Key Concept

Calculating the volume of a cylinder using the radius and height.

Applied Geometry

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