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

Question 1Question →

A city is installing a new fence around a rectangular dog park. The park has a length of $80$ feet and a width of $60$ feet. How many feet of fencing are required to completely enclose the park?

Show answer & explanation

Answer: 280

Answer

280

The fencing must cover the entire outer edge of the rectangular park. The perimeter is found by adding the length and width together, then multiplying the sum by 2 to account for all four sides. Thus, 2 × (80 + 60) = 280 feet.

Step-by-Step Solution

1

Identify the given dimensions of the rectangular park.

Length = 80 feet, Width = 60 feet.

To determine the values needed for the perimeter formula.

2

Apply the formula for the perimeter of a rectangle.

Perimeter = 2 × (80 + 60)

The total fencing required is equal to the perimeter of the rectangle.

3

Perform the addition inside the parentheses.

80 + 60 = 140

Following the order of operations.

4

Multiply the sum by 2.

2 × 140 = 280

To account for all four sides of the rectangular boundary.

Key Concept

The perimeter of a rectangle is the total distance around the outside edge, calculated using the formula 2 × (Length + Width).

Question 2Question →

A homeowner is buying new carpet for a rectangular bedroom that measures $15$ feet long and $12$ feet wide. If the carpet costs $5$ dollars per square foot, what is the total cost, in dollars, of the carpet?

900

270

90

36

Show answer & explanation

Answer: 900

Answer

The total cost of the carpet is

900

dollars.

The total area of the floor is found by multiplying the length by the width (

15 \times 12 = 180

square feet). Since the carpet costs

5

dollars per square foot, the total cost is the area multiplied by the unit price (

180 \times 5 = 900

).

Step-by-Step Solution

1

Calculate the total area of the rectangular bedroom.

15 \times 12 = 180

square feet

The cost of the carpet is based on the floor area it covers, which is found by multiplying length by width.

2

Calculate the total cost by multiplying the area by the price per square foot.

180 \times 5 = 900

Each square foot costs

5

dollars, so the total cost is the area times the unit rate.

Key Concept

Calculating the area of a rectangle and using it to find a total cost based on a unit rate.

Question 3Question →

A homeowner is planning to install new carpet in a rectangular living room that measures $18\text{ feet}$ long by $15\text{ feet}$ wide. Inside the room, there is a built-in rectangular closet measuring $9\text{ feet}$ long by $3\text{ feet}$ wide that will not be carpeted. If the selected carpet costs $\$20$ per square yard, what is the total cost of the carpet needed for the room?

$1,620

$440

$540

$4,860

Show answer & explanation

Answer: $540

Answer

$540

The correct answer accurately calculates the net area in square feet (

270 - 27 = 243\text{ sq ft}

), converts it to square yards by dividing by 9 (

27\text{ sq yd}

), and then multiplies by the rate of

\$20

per square yard to find the total cost of

\$540

.

Step-by-Step Solution

1

Calculate the total area of the living room.

18 \times 15 = 270\text{ sq ft}

You must first determine the total floor space before accounting for the uncarpeted closet area.

2

Calculate the area of the uncarpeted closet.

9 \times 3 = 27\text{ sq ft}

The area of the closet needs to be identified so it can be excluded from the total carpet cost.

3

Subtract the closet area from the total room area.

270 - 27 = 243\text{ sq ft}

This provides the net floor area that requires carpeting.

4

Convert the net area from square feet to square yards.

243 \div 9 = 27\text{ sq yd}

The carpet is priced per square yard, and there are

9\text{ square feet}

in

1\text{ square yard}

(

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

).

5

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

27 \times 20 = \$540

Multiplying the required amount of carpet by its unit price gives the final total cost.

Key Concept

Calculating subtractive areas in applied contexts and converting between square units.

Estimated Time:1m 30s

Question 4Question →

A rectangular community hall measures $60 \text{ feet}$ long and $45 \text{ feet}$ wide. A rectangular stage measuring $15 \text{ feet}$ long and $12 \text{ feet}$ wide is located inside the hall and will not be carpeted. If commercial carpet costs $\$24$ per square yard, what is the total cost to carpet the remainder of the hall?

$1,248

$6,720

$20,160

$60,480

Show answer & explanation

Answer: $6,720

Answer

The total cost to carpet the remainder of the hall is $6,720.

The correct approach requires finding the net area to be carpeted in square feet (

2,700 - 180 = 2,520

), converting it to square yards by dividing by 9 (

280 \text{ sq yd}

), and multiplying by the cost per square yard (

24) to reach the final cost of

6,720.

Step-by-Step Solution

1

Calculate the total area of the community hall.

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

To find the gross area before removing the uncarpeted section.

2

Calculate the area of the uncarpeted stage.

15 \times 12 = 180 \text{ square feet}

To determine the exact area that will not receive carpet.

3

Subtract the stage area from the total hall area.

2,700 - 180 = 2,520 \text{ square feet}

To find the net area that requires carpeting.

4

Convert the net area from square feet to square yards.

2,520 \div 9 = 280 \text{ square yards}

Because the carpet cost is given per square yard, and $1 \text{ square yard} = 3 \text{ feet} \times 3 \text{ feet} = 9 \text{ square feet}.

5

Calculate the total cost of the carpet.

280 \times 24 = \$6,720

To find the final price by multiplying the area in square yards by the cost per square yard.

Key Concept

Calculating the net area by subtracting a smaller rectangular region from a larger one, and converting between square units (

1 \text{ sq yd} = 9 \text{ sq ft}

) before calculating total cost.

Question 5Question →

A contractor is installing new flooring in a rectangular hall that measures $24\text{ feet}$ by $36\text{ feet}$ . Inside the hall, there is a rectangular stage measuring $12\text{ feet}$ by $9\text{ feet}$ that will not receive new flooring. The flooring material is sold in boxes that each cover $2\text{ square yards}$ and cost $\$45.50$ per box. Assuming no waste, what is the total cost of the boxes needed to complete the job?

\$591.50

\$1,911.00

\$3,822.00

\$5,733.00

Show answer & explanation

Answer: $\$1,911.00$

Answer

\$1,911.00

The correct answer is found by calculating the net surface area and converting the units properly. The hall's area is 864 square feet and the stage's area is 108 square feet, leaving a net area of 756 square feet. Because 1 square yard equals 9 square feet, this converts to 84 square yards. Dividing by 2 square yards per box gives 42 boxes. At

45.50 each, the total cost is

1,911.00.

Step-by-Step Solution

1

Calculate the total area of the hall in square feet.

24 \times 36 = 864\text{ sq ft}

To find the maximum possible flooring area before accounting for the unfloored stage.

2

Calculate the area of the stage in square feet.

12 \times 9 = 108\text{ sq ft}

The stage will not receive new flooring, so its area must be determined for subtraction.

3

Calculate the net area to be floored.

864 - 108 = 756\text{ sq ft}

Subtracting the stage area from the hall area gives the actual surface area that requires new flooring.

4

Convert the net area from square feet to square yards.

756 \div 9 = 84\text{ sq yd}

The flooring boxes measure coverage in square yards. There are

9\text{ sq ft}

in

1\text{ sq yd}

(

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

).

5

Determine the number of boxes needed.

84 \div 2 = 42\text{ boxes}

Each box covers

2\text{ sq yd}

, so dividing the total square yards by the coverage per box yields the required number of boxes.

6

Calculate the total cost of the boxes.

42 \times \$45.50 = \$1,911.00

Multiplying the total number of boxes by the price per box gives the final project cost.

Key Concept

Calculating net rectangular area with cutouts, converting between square units, and applying rates to find total material cost.

Question 6Question →

A hotel manager is ordering custom acoustic paneling for a large rectangular conference room wall. The wall measures $36 \text{ feet}$ in length and $15 \text{ feet}$ in height. The wall contains $3$ identical rectangular windows, each measuring $6 \text{ feet}$ wide and $4 \text{ feet}$ high, which will not be covered by the paneling. If the acoustic paneling costs $\$45$ per square yard, what is the total cost of the paneling needed for this wall?

\$1,530

\$2,340

\$7,020

\$21,060

Show answer & explanation

Answer: $\$2,340$

Answer

The total cost of the acoustic paneling needed is

\$2,340

.

The correct answer is found by first determining the gross area of the wall (

36 \times 15 = 540 \text{ sq ft}

) and subtracting the area of the three windows (

3 \times 24 = 72 \text{ sq ft}

) to get a net coverage area of

468 \text{ sq ft}

. Because the paneling is sold by the square yard, the square footage must be converted by dividing by

9

(since

1 \text{ sq yd} = 9 \text{ sq ft}

). This gives

52 \text{ sq yd}

. Multiplying this by the price of

\$45

per square yard yields a total cost of

\$2,340

.

Step-by-Step Solution

1

Calculate the total area of the rectangular wall in square feet.

36 \times 15 = 540 \text{ sq ft}

The gross area is needed before accounting for the cutouts.

2

Calculate the combined area of the three identical windows.

3 \times (6 \times 4) = 72 \text{ sq ft}

The windows will not be covered by paneling, so their total area must be subtracted.

3

Subtract the window area from the total wall area to find the net area to be covered.

540 - 72 = 468 \text{ sq ft}

This provides the exact amount of paneling needed in square feet.

4

Convert the net area from square feet to square yards.

468 \div 9 = 52 \text{ sq yd}

Since the cost is given per square yard, the area must match this unit. There are

3 \times 3 = 9 \text{ sq ft}

in a square yard.

5

Multiply the required area in square yards by the cost per square yard.

52 \times 45 = \$2,340

This yields the total cost for the materials.

Key Concept

Calculating subtractive rectangular area and converting square units before applying a cost rate.

Question 7Question →

A city planner is designing a rectangular public square that measures $60 \text{ yards}$ by $40 \text{ yards}$ . The entire square will be paved with brick, except for a central rectangular fountain that measures $10 \text{ yards}$ by $8 \text{ yards}$ . If the paving bricks cost $\$2.50$ per square foot, what is the total cost of the bricks needed for the square in dollars?

Show answer & explanation

Answer: 52200

Answer

52200

The correct answer is

52200 because the net area of the square to be paved is

2,320

square yards (

2400 \text{ sq yd total} - 80 \text{ sq yd fountain}

). Since

1

square yard equals

9

square feet, this converts to

20,880

square feet. Multiplying the total square footage by the brick cost of

\

2.50

per square foot yields a total cost of

\$52,200

.

Step-by-Step Solution

1

Calculate the total area of the public square in square yards.

2400 \text{ sq yd}

The total area is the product of its length and width (

60 \text{ yards} \times 40 \text{ yards}

).

2

Calculate the area of the central fountain in square yards.

80 \text{ sq yd}

The area of the fountain is the product of its length and width (

10 \text{ yards} \times 8 \text{ yards}

).

3

Subtract the fountain's area from the total area to find the net paved area in square yards.

2320 \text{ sq yd}

The fountain will not be paved, so its area must be removed from the total square yardage (

2400 - 80

).

4

Convert the net area from square yards to square feet.

20880 \text{ sq ft}

The cost of the bricks is given per square foot. Since

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

,

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

. Therefore, we multiply the area in square yards by

9

(

2320 \times 9 = 20880

).

5

Calculate the total cost by multiplying the net area in square feet by the price per square foot.

52200

Multiplying the total square footage (

20880

) by the cost per square foot (

\$2.50

) yields the final cost in dollars.

Key Concept

Subtractive area calculations and square unit conversions

Question 8Question →

A contractor is preparing to apply sealant to a new rectangular patio that measures $14$ yards in length and $5$ yards in width. If the sealant costs $3$ dollars per square yard, what is the total cost, in dollars, to seal the entire patio?

Show answer & explanation

Answer: 210

Answer

210

To find the total cost, you must first determine the area of the patio by multiplying the length (

14

yards) by the width (

5

yards), which equals

70

square yards. Then, multiply this area by the cost per square yard (

3

dollars) to get the total cost of

210

dollars.

Step-by-Step Solution

1

Calculate the area of the rectangular patio.

Area =

14 \times 5 = 70

square yards

The cost of the sealant is based on the surface area of the patio.

2

Calculate the total cost to seal the patio.

Total cost =

70 \times 3 = 210

dollars

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

Key Concept

Calculating rectangular area and applying a unit rate.

Question 9Question →

A city planner is designing a rectangular park that features a central grassy area measuring $150 \text{ feet}$ long and $90 \text{ feet}$ wide. A uniform concrete walkway that is $6 \text{ feet}$ wide will be constructed around the entire outer perimeter of the grassy area. If the concrete paving costs $\$12.50$ per square yard, what is the total cost, in dollars, of the paving material required for the walkway?

Show answer & explanation

Answer: 4200

Answer

4200

The correct cost is found by determining the total area of the walkway by subtracting the inner grass area from the outer total area (

16,524 - 13,500 = 3,024 \text{ sq ft}

), converting to square yards by dividing by 9 (

336 \text{ sq yd}

), and multiplying by the price of

\$12.50

per square yard.

Step-by-Step Solution

1

Determine the outer dimensions of the total area.

Outer length = 162 ft, Outer width = 102 ft.

The walkway is 6 feet wide and borders all sides. Therefore, 6 feet must be added to both ends of the length (

150 + 6 + 6 = 162

) and both sides of the width (

90 + 6 + 6 = 102

).

2

Calculate the area of the outer rectangle.

16,524 sq ft

Area is length multiplied by width:

162 \times 102 = 16,524

.

3

Calculate the area of the inner grassy rectangle.

13,500 sq ft

Using the given inner dimensions:

150 \times 90 = 13,500

.

4

Find the area of the walkway.

3,024 sq ft

Subtract the inner area from the outer area:

16,524 - 13,500 = 3,024

.

5

Convert the walkway area from square feet to square yards.

336 sq yd

There are

3 \times 3 = 9

square feet in one square yard. Divide the area by 9:

3,024 \div 9 = 336

.

6

Calculate the total cost.

$4,200

Multiply the area in square yards by the cost per square yard:

336 \times 12.50 = 4,200

.

Key Concept

Calculating subtractive rectangular area and converting square units for cost analysis.

Question 10Question →

A commercial painting crew is hired to paint the four exterior walls of a rectangular warehouse. The warehouse has a length of $120\text{ ft}$ , a width of $80\text{ ft}$ , and a height of $24\text{ ft}$ . The building features $2$ identical loading doors, each measuring $16\text{ ft}$ wide by $15\text{ ft}$ high, and $4$ identical windows, each measuring $6\text{ ft}$ wide by $5\text{ ft}$ high. The doors and windows will not be painted. The crew must apply exactly $2$ coats of paint to all paintable exterior wall surfaces. If one gallon of paint covers exactly $400\text{ square feet}$ , how many total gallons of paint are required for this project?

Show answer & explanation

Answer: 45

Answer

45

The total area of the four exterior walls is 9,600 sq ft. The doors and windows have a combined area of 600 sq ft, leaving a net paintable area of 9,000 sq ft per coat. Since two coats are required, the total area to cover is 18,000 sq ft. With each gallon covering 400 sq ft, dividing 18,000 by 400 gives exactly 45 gallons.

Step-by-Step Solution

1

Calculate the gross area of the four exterior walls.

9,600 sq ft

To find the total surface area before accounting for doors and windows.

2

Calculate the total area of the unpainted cutouts.

600 sq ft

The doors and windows will not be painted and must be subtracted.

3

Determine the net paintable area for a single coat.

9,000 sq ft

Subtracting the cutout area from the gross area gives the actual area to paint.

4

Calculate the total area to be covered for two coats.

18,000 sq ft

The problem specifies that exactly two coats of paint must be applied.

5

Determine the total gallons of paint required.

45

Divide the total area to be painted by the coverage rate of 400 sq ft per gallon.

Key Concept

Calculating net rectangular area by subtracting cutouts and applying a rate for multiple layers.

Question 11Question →

A local sports center is installing a new rectangular practice mat for gymnastics. The designated floor space is $40$ feet long and $20$ feet wide. If the specialized foam matting costs $\$5$ per square foot, what is the total cost to cover this floor space?

\$400

\$600

\$800

\$4,000

Show answer & explanation

Answer: $\$4,000$

Answer

The total cost to cover the floor space is

\$4,000

.

To find the total cost, we first need to determine the total amount of matting required by calculating the area of the rectangular space. The area is found by multiplying the length by the width:

40 \times 20 = 800

square feet. Then, we multiply this total area by the cost of the matting per square foot:

800 \text{ square feet} \times \$5 \text{ per square foot} = \$4,000

.

Step-by-Step Solution

1

Calculate the area of the rectangular floor space.

40 \text{ feet} \times 20 \text{ feet} = 800 \text{ square feet}

.

The cost of the matting is priced per square foot, so you must first determine the total area of the floor space.

2

Calculate the total cost by multiplying the area by the price per square foot.

800 \text{ square feet} \times \$5/\text{square foot} = \$4,000

.

Multiplying the total square footage by the cost per unit of area gives the total expense.

Key Concept

Calculating the area of a rectangle to determine total material costs based on a unit rate.

Question 12Question →

A landscaper is installing sod in a rectangular backyard that is $45$ feet long and $30$ feet wide. If the sod costs $\$6.00$ per square yard, what is the total cost of the sod for the backyard?

\$300

\$900

\$2,700

\$8,100

Show answer & explanation

Answer: $\$900$

Answer

The total cost of the sod for the backyard is

\$900

.

The correct answer is found by determining the area in square yards and multiplying by the unit price. Converting the dimensions first gives

15

yards by

10

yards, yielding an area of

150

square yards. Multiplying

150

by

\$6.00

gives a total cost of

\$900

.

Step-by-Step Solution

1

Convert the dimensions of the rectangular backyard from feet to yards.

Length =

45 \div 3 = 15

yards; Width =

30 \div 3 = 10

yards.

Since the cost is given per square yard, it is most efficient to calculate the area in square yards by first converting the linear dimensions.

2

Calculate the area of the backyard in square yards.

15 \text{ yards} \times 10 \text{ yards} = 150 \text{ square yards}

.

The area of a rectangle is found by multiplying its length by its width.

3

Calculate the total cost of the sod.

150 \text{ square yards} \times \$6.00/\text{square yard} = \$900

.

Multiplying the total area in square yards by the cost per square yard gives the final total cost.

Key Concept

Calculating rectangular area and applying appropriate unit conversions to estimate total cost.

Alternative Method

You can also calculate the area in square feet first (

45 \times 30 = 1,350

sq ft), then divide by

9

to convert to square yards (

1,350 \div 9 = 150

sq yds), and finally multiply by the cost per square yard.

Estimated Time:1m 30s

Question 13Question →

A facility manager is replacing the flooring in a rectangular open-plan office that measures $60\text{ ft}$ in length and $46\text{ ft}$ in width. The office contains two identical rectangular utility columns, each measuring $6\text{ ft}$ by $5\text{ ft}$ , which will not be covered with the new flooring. If the new flooring material costs $\$22.00$ per square yard, what is the total cost of the flooring material required for the office in dollars?

Show answer & explanation

Answer: 6600

Answer

6600

The correct calculation correctly subtracts the area of the utility columns to find the net area in square feet, correctly converts this area to square yards by dividing by 9, and then multiplies by the price per square yard to find the final total cost.

Step-by-Step Solution

1

Calculate the gross area of the office.

2760 sq ft

The total rectangular area without accounting for the utility columns is

60 \times 46

.

2

Calculate the total area of the two utility columns.

60 sq ft

Each column is

6 \times 5 = 30\text{ sq ft}

, and there are two of them, so

30 \times 2 = 60\text{ sq ft}

.

3

Subtract the column area from the gross area.

2700 sq ft

This provides the net area that requires new flooring.

4

Convert the net area from square feet to square yards.

300 sq yd

The flooring cost is given per square yard, and there are

9\text{ sq ft}

in

1\text{ sq yd}

. Therefore,

2700 \div 9 = 300

.

5

Calculate the total cost of the flooring material.

6600

Multiply the area in square yards by the cost per square yard:

300 \times 22 = 6600

.

Key Concept

Calculating subtractive rectangular area and applying square unit conversions to determine total cost.

Question 14Question →

A city is designing a circular park with a diameter of $60$ meters. In the center of the park, there will be a circular concrete plaza with a diameter of $20$ meters. The rest of the park will be covered in grass. If the cost to plant grass is $\$0.50$ per square meter, what will be the total cost to plant the grass, expressed in terms of $\pi$ ?

\$20\pi

\$400\pi

\$800\pi

\$1,600\pi

Show answer & explanation

Answer: $\$400\pi$

Answer

The correct total cost is

\$400\pi

.

The correct answer is found by first calculating the area of the entire circular park and subtracting the area of the concrete plaza. Using the formula

Area = \pi r^2

, the park's area is

\pi(30)^2 = 900\pi

and the plaza's area is

\pi(10)^2 = 100\pi

. The difference is

800\pi

square meters. Multiplying this by the cost of

\$0.50

per square meter results in a total cost of

\$400\pi

.

Step-by-Step Solution

1

Find the radii of the entire park and the central plaza.

Park radius =

30

m. Plaza radius =

10

m.

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

2

Calculate the total area of the park and the area of the plaza in terms of

\pi

.

Total Area =

\pi(30)^2 = 900\pi

. Plaza Area =

\pi(10)^2 = 100\pi

.

We must determine the individual areas to find the space remaining for the grass.

3

Subtract the plaza area from the total park area to find the area to be covered in grass.

Grass Area =

900\pi - 100\pi = 800\pi

square meters.

The grass will cover the portion of the park that is not occupied by the concrete plaza.

4

Multiply the grass area by the unit cost to plant the grass.

Total Cost =

800\pi \times \$0.50 = \$400\pi

.

The final question asks for the total cost, which is the area multiplied by the price per square meter.

Key Concept

Calculating the area of an annulus (a ring shape) by subtracting the area of an inner circle from an outer circle, and applying unit rates.

Estimated Time:2m 0s

Question 15Question →

An agricultural company is installing a weed-blocking fabric on the dirt floor of a rectangular greenhouse that is $120 \text{ feet}$ long and $45 \text{ feet}$ wide. Inside the greenhouse, there are three identical rectangular planting beds, each measuring $30 \text{ feet}$ by $12 \text{ feet}$ , which will not be covered by the fabric. The fabric is sold in rolls that cover $15 \text{ square yards}$ each, and each roll costs $\$45$ . Assuming partial rolls cannot be purchased and the company must buy whole rolls, what is the total cost of the fabric rolls needed to cover the designated floor area?

\$990

\$4,320

\$1,440

\$12,960

Show answer & explanation

Answer: $\$1,440$

Answer

The total cost of the fabric rolls needed is

\$1,440

.

The correct total cost is found by determining the subtractive area in square feet (

5,400 - 1,080 = 4,320

), converting that area to square yards by dividing by

9

(

480 \text{ sq yd}

), determining the number of

15

-square-yard rolls needed (

32

), and multiplying by the price per roll (

\$45

).

Step-by-Step Solution

1

Calculate the total area of the greenhouse floor.

120 \text{ feet} \times 45 \text{ feet} = 5,400 \text{ sq ft}

This establishes the maximum possible area to be covered before accounting for the exclusions.

2

Calculate the total area of the three planting beds.

3 \times (30 \text{ feet} \times 12 \text{ feet}) = 3 \times 360 \text{ sq ft} = 1,080 \text{ sq ft}

The planting beds are not getting covered, so their combined area must be subtracted.

3

Subtract the area of the planting beds from the total greenhouse area.

5,400 \text{ sq ft} - 1,080 \text{ sq ft} = 4,320 \text{ sq ft}

This provides the net area of the floor that requires the weed-blocking fabric.

4

Convert the net area from square feet to square yards.

4,320 \text{ sq ft} \div 9 \text{ sq ft/sq yd} = 480 \text{ sq yd}

The fabric is sold based on coverage in square yards, so the area units must match. There are

9

square feet in one square yard (

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

).

5

Determine the number of rolls needed.

480 \text{ sq yd} \div 15 \text{ sq yd/roll} = 32 \text{ rolls}

Dividing the total square yards by the coverage per roll gives the required number of rolls.

6

Calculate the total cost of the rolls.

32 \text{ rolls} \times \$45/\text{roll} = \$1,440

Multiplying the needed number of rolls by the price per roll provides the final cost.

Key Concept

Rectangular Area Applications

Estimated Time:2m 30s

Question 16Question →

A military supply vehicle is equipped with tires that have a radius of $14$ inches. How many complete revolutions does each tire make if the vehicle travels a straight-line distance of $1.5$ miles? (Use $\pi = \frac{22}{7}$ and $1 \text{ mile} = 5,280 \text{ feet}$ .)

Show answer & explanation

Answer: 1080

Answer

1080

The total distance must be converted into inches (

1.5 \text{ miles} \times 5,280 \text{ ft/mile} \times 12 \text{ in/ft} = 95,040 \text{ inches}

). The tire's circumference (

2 \times \frac{22}{7} \times 14 = 88 \text{ inches}

) represents one revolution. Dividing the total distance by the circumference (

95,040 \div 88

) gives exactly

1,080

complete revolutions.

Step-by-Step Solution

1

Convert the total travel distance from miles to feet.

1.5 \times 5,280 = 7,920

feet

The distance must be in the same units as the tire measurements. It is easier to step down to feet first.

2

Convert the distance from feet to inches.

7,920 \times 12 = 95,040

inches

Since the tire radius is in inches, the total distance must also be in inches to calculate revolutions accurately.

3

Calculate the circumference of the tire.

C = 2 \times \pi \times r = 2 \times \frac{22}{7} \times 14 = 88

inches

One revolution of the tire equals its circumference, representing the distance traveled in a single rotation.

4

Calculate the total number of complete revolutions.

95,040 \div 88 = 1,080

revolutions

Dividing the total distance by the distance per revolution yields the total number of revolutions.

Key Concept

Calculating distance traveled using wheel revolutions requires relating linear distance to circular circumference, while ensuring all units of measurement are consistent.

Estimated Time:2m 0s

Question 17Question →

A business owner is paving a rectangular parking lot that measures $40\text{ feet}$ long and $30\text{ feet}$ wide. If the paving costs $\$5$ per square foot, what is the total cost of paving the lot in dollars?

Show answer & explanation

Answer: 6000

Answer

6000

The parking lot is a rectangle, so its area is found by multiplying the length by the width (

40 \times 30 = 1,200\text{ sq ft}

). Since the paving costs

\$5

for every square foot, the total cost is the area multiplied by the rate (

1,200 \times 5 = 6,000

). The final cost is

\$6,000

.

Step-by-Step Solution

1

Calculate the total area of the rectangular parking lot.

1,200 square feet

The total area is required to know how many square feet need to be paved.

2

Multiply the calculated area by the cost per square foot.

6,000

Each of the 1,200 square feet costs $5 to pave, so multiplying them gives the total cost.

Key Concept

Calculating the area of a rectangle and applying it to find the total cost of materials or services.

Estimated Time:45s

Question 18Question →

A community center is installing a protective rubber surface on a rectangular play area. The play area is $20\text{ feet}$ long and $15\text{ feet}$ wide. If the rubber surface material costs $\$5.00$ per square foot, what is the total cost of the material?

\$150

\$300

\$350

\$1,500

Show answer & explanation

Answer: \$1,500

Answer

\$1,500

To find the total cost, first calculate the area of the rectangular space by multiplying the length by the width (

20 \times 15 = 300\text{ square feet}

). Then, multiply the total square footage by the cost per square foot (

300 \times 5 = 1,500

).

Step-by-Step Solution

1

Calculate the total area of the rectangular play area.

\text{Area} = 20\text{ feet} \times 15\text{ feet} = 300\text{ square feet}

The cost of the material is based on the square footage required to cover the surface, so the area must be determined first.

2

Multiply the total area by the cost per square foot to find the total cost.

\text{Total Cost} = 300\text{ sq ft} \times \$5.00/\text{sq ft} = \$1,500

This determines the final price by scaling the calculated area by the given unit cost.

Key Concept

Calculating the area of a rectangle and applying unit rates to find a total cost.

Question 19Question →

A roofing crew is applying a waterproof liquid coating to a flat, rectangular building roof that is $90\text{ feet}$ long and $60\text{ feet}$ wide. If the coating material costs $\$12$ per square yard, what is the total cost to coat the entire roof?

\$1,200

\$7,200

\$21,600

\$64,800

Show answer & explanation

Answer: $\$7,200$

Answer

The total cost to coat the entire roof is

\$7,200

.

To find the total cost, the area must be calculated in square yards because the price is given per square yard. Since

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

, the dimensions of the roof are

30\text{ yards}

by

20\text{ yards}

. The area is

30 \times 20 = 600\text{ square yards}

. Multiplying this area by the

\$12

cost per square yard yields

\$7,200

.

Step-by-Step Solution

1

Convert the length and width of the roof from feet to yards.

Length =

30\text{ yards}

, Width =

20\text{ yards}

.

The cost of the coating is given per square yard, so it is most efficient to work with linear dimensions in yards (

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

).

2

Calculate the total area of the roof in square yards.

30\text{ yards} \times 20\text{ yards} = 600\text{ square yards}

.

The area of a rectangle is found by multiplying its length by its width.

3

Multiply the total area by the cost per square yard.

600\text{ sq yd} \times \$12\text{/sq yd} = \$7,200

.

This determines the final material cost required to cover the entire area.

Key Concept

Calculating rectangular area and applying proper unit conversions between square feet and square yards to determine total cost.

Question 20Question →

A homeowner is applying a protective sealant to a rectangular wooden patio that completely surrounds a rectangular swimming pool. The overall dimensions of the patio and pool together are $40\text{ feet}$ long and $30\text{ feet}$ wide. The swimming pool inside measures $25\text{ feet}$ long and $15\text{ feet}$ wide. One gallon of sealant covers exactly $150\text{ square feet}$ of wood and costs $\$28$ . If the homeowner must purchase whole gallons of sealant, what is the total cost, in dollars, of the sealant required for the wooden patio?

Show answer & explanation

Answer: 168

Answer

The total cost of the sealant is $168.

The net area of the patio is found by subtracting the pool area from the total area (

1200 - 375 = 825\text{ sq ft}

). Dividing this by the coverage rate (

825 \div 150 = 5.5

) shows how many gallons are needed. Since partial gallons are not sold, this rounds up to

6\text{ gallons}

. At

\$28

per gallon, the cost is

6 \times 28 = \$168

.

Step-by-Step Solution

1

Calculate the total combined area of the patio and the pool.

1200\text{ square feet}

To find the gross area before removing the inner section, multiply the overall length by the width (

40 \times 30

).

2

Calculate the area of the swimming pool.

375\text{ square feet}

To find the area that does not require sealant, multiply the pool's length by its width (

25 \times 15

).

3

Subtract the pool area from the total area.

825\text{ square feet}

To find the net area of the wooden patio that actually needs sealant (

1200 - 375

).

4

Divide the net area by the coverage of one gallon of sealant.

5.5\text{ gallons}

To determine the exact volume of sealant required (

825 \div 150

).

5

Round up to the next whole gallon.

6\text{ gallons}

Because the homeowner must purchase whole gallons, any fraction requires buying an additional full gallon.

6

Multiply the number of gallons by the cost per gallon.

\$168

To calculate the final total cost (

6 \times 28

).

Key Concept

Calculating net rectangular area by subtracting an inner rectangle from an outer boundary, and applying real-world constraints such as rounding up to whole purchasing units.

Estimated Time:1m 30s

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