Applied Geometry Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 3

Question 41Question →

An industrial steel cable is wrapped around a cylindrical hoisting drum that has a diameter of $2.5$ feet. If the drum must complete exactly $40$ full rotations to completely unwind the cable, what is the total length of the cable? (Use $\pi = 3.14$ )

7.85

feet

196.25

feet

314

feet

628

feet

Show answer & explanation

Answer: $314$ feet

Answer

The total length of the cable is

314

feet.

The length of cable unwound in one rotation is exactly equal to the circumference of the drum. Using the formula

C = \pi \times d

, the circumference is

3.14 \times 2.5 = 7.85

feet. Since the drum rotates

40

times, the total cable length is

7.85 \text{ feet/rotation} \times 40 \text{ rotations} = 314

feet.

Step-by-Step Solution

1

Calculate the circumference of the cylindrical drum.

C = \pi \times d = 3.14 \times 2.5 = 7.85

feet.

One full rotation of the drum unwinds an amount of cable equal to the drum's circumference.

2

Multiply the circumference by the number of rotations to find the total length.

7.85 \times 40 = 314

feet.

The drum completes

40

rotations, meaning

40

lengths of the circumference are unwound.

Key Concept

Distance from Rotations (Circumference)

Question 42Question →

A cylindrical farm silo is built to store corn. The interior of the silo has a radius of $7\text{ feet}$ and a height of $20\text{ feet}$ . Using $\frac{22}{7}$ for $\pi$ , what is the total volume of the silo in cubic feet?

Show answer & explanation

Answer: 3080

Answer

The total volume of the silo is 3080 cubic feet.

The correct answer is calculated by applying the volume formula for a cylinder,

V = \pi r^2 h

. Substituting

7

for the radius and

20

for the height, along with

\frac{22}{7}

for

\pi

, results in

3080\text{ cubic feet}

.

Step-by-Step Solution

1

Recall the volume formula

V = \pi r^2 h

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

2

Substitute the given values

V = \frac{22}{7} \times (7)^2 \times 20

We plug in

r = 7

,

h = 20

, and

\pi = \frac{22}{7}

as provided in the question.

3

Square the radius

V = \frac{22}{7} \times 49 \times 20

Following the order of operations, exponents must be calculated before multiplication.

4

Multiply and simplify

V = 22 \times 7 \times 20 = 3080

Simplifying the fraction

\frac{22}{7} \times 49

leaves

22 \times 7 = 154

. Multiplying

154

by

20

yields

3080

.

Key Concept

Calculating the volume of a cylinder

Question 43Question →

A machinist is fabricating a circular metal part with a diameter of $12$ inches. To allow for bolts, four identical circular holes, each with a radius of $2$ inches, are cut out of the part. What is the area of the remaining metal in square inches? (Use $\pi = 3.14$ )

Show answer & explanation

Answer: 62.8

Answer

62.8

The remaining area is found by subtracting the total area of the four smaller circular holes from the area of the large circular part. The large part has a radius of 6 inches (half of the 12-inch diameter), giving an area of 113.04 square inches. Each of the four holes has a radius of 2 inches, yielding an area of 12.56 square inches per hole, or 50.24 square inches total. Subtracting 50.24 from 113.04 leaves 62.8 square inches.

Step-by-Step Solution

1

Find the radius of the large circular part.

Radius = 12 / 2 = 6 inches.

The area formula for a circle requires the radius, but the diameter was given.

2

Calculate the total area of the large circular part.

Area = 3.14 * 6^2 = 113.04 square inches.

This represents the total amount of metal before any holes are cut.

3

Calculate the area of one small bolt hole.

Area = 3.14 * 2^2 = 12.56 square inches.

Each hole is a smaller circle removed from the part.

4

Find the total area of all four bolt holes.

Total hole area = 12.56 * 4 = 50.24 square inches.

There are four identical holes that need to be accounted for.

5

Subtract the total hole area from the large part's area.

Remaining area = 113.04 - 50.24 = 62.8 square inches.

Subtracting the removed material gives the final remaining area of the metal part.

Key Concept

Calculating and subtracting circular areas

Question 44Question →

A homeowner is enclosing a rectangular garden with a new fence. The garden has a length of $5$ yards and a width of $8$ feet. The homeowner will leave a $36$ -inch wide opening along one of the sides for a wooden gate. If the fencing material costs $\$14.50$ per linear foot, what is the total cost of the fencing material required?

\$333.50

\$623.50

\$1,740.00

\$6,235.00

Show answer & explanation

Answer: $\$623.50$

Answer

\$623.50

To find the correct total cost, all dimensions must first be converted to feet (

5

yards

= 15

feet,

36

inches

= 3

feet). The perimeter is found by adding all four sides (

15 + 15 + 8 + 8 = 46

feet), then subtracting the

3

-foot gate opening to determine that exactly

43

feet of fencing is needed. Multiplying

43

feet by the

\$14.50

cost per foot yields the correct total of

\$623.50

.

Step-by-Step Solution

1

Convert all given measurements to a single common unit (feet) since the cost is given per linear foot.

Length:

5 \text{ yards} = 15 \text{ feet}

. Gate width:

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

.

Measurements must be in the same units as the pricing rate to calculate accurate costs.

2

Calculate the total perimeter of the rectangular garden.

P = 2 \times (\text{Length} + \text{Width}) = 2 \times (15 + 8) = 2 \times 23 = 46 \text{ feet}

.

The perimeter represents the total outline of the garden before accounting for the gate.

3

Subtract the width of the gate opening from the total perimeter.

46 \text{ feet} - 3 \text{ feet} = 43 \text{ feet}

.

Fencing material is not needed across the opening where the wooden gate will be installed.

4

Multiply the required length of fencing by the cost per linear foot.

43 \text{ feet} \times \$14.50/\text{foot} = \$623.50

.

This provides the final total monetary cost for the required material.

Key Concept

Perimeter and Unit Conversion

Question 45Question →

A wooden ramp used for moving equipment is 17 feet long. If the horizontal distance from the base of the ramp to the building is 15 feet, what is the vertical height of the top of the ramp?

2 feet

8 feet

32 feet

64 feet

Show answer & explanation

Answer: 8 feet

Answer

8 feet

The scenario describes a right triangle where the ramp is the hypotenuse, and the distance to the building is one of the legs. Using the Pythagorean theorem,

a^2 + b^2 = c^2

. Plugging in the knowns gives

15^2 + b^2 = 17^2

, which evaluates to

225 + b^2 = 289

. Subtracting 225 from both sides yields

b^2 = 64

. The square root of 64 is 8, so the ramp reaches a vertical height of 8 feet.

Step-by-Step Solution

1

Identify the known and unknown sides of the right triangle formed by the ramp, the ground, and the building.

The hypotenuse (ramp) is

c = 17

, the horizontal leg (ground) is

a = 15

, and the vertical leg (height) is

b

.

To determine which part of the Pythagorean theorem formula (

a^2 + b^2 = c^2

) needs to be solved.

2

Substitute the known values into the Pythagorean theorem and square them.

15^2 + b^2 = 17^2

becomes

225 + b^2 = 289

.

To set up the equation for isolating the unknown variable.

3

Subtract 225 from 289 to isolate

b^2

, then take the square root.

b^2 = 289 - 225 = 64

. Taking the square root gives

b = \sqrt{64} = 8

.

To find the final linear measurement of the vertical height.

Key Concept

Applying the Pythagorean theorem to find a missing leg of a right triangle.

Estimated Time:45s

Question 46Question →

A city park features a circular fountain with a diameter of $20$ feet. The city plans to pour a new concrete walkway that is exactly $2$ feet wide completely surrounding the fountain. If the concrete costs $\$5.00$ per square foot, what is the total cost of the concrete required for the new walkway? (Use $\pi = 3.14$ )

\$31.40

\$62.80

\$690.80

\$1,318.80

Show answer & explanation

Answer: \$690.80

Answer

The total cost of the concrete required for the new walkway is \$690.80.

The correct answer is found by calculating the area of the walkway and multiplying it by the cost per square foot. First, find the radius of the fountain by halving the diameter (

20 \div 2 = 10

feet). The outer radius includes both the fountain and the walkway (

10 + 2 = 12

feet). The area of the walkway is the outer area minus the inner area:

(3.14 \times 12^2) - (3.14 \times 10^2) = 452.16 - 314 = 138.16

square feet. Finally, multiplying this area by the cost of

\$5.00

per square foot gives

\$690.80

.

Step-by-Step Solution

1

Determine the inner and outer radii of the circular areas.

Inner radius =

10

feet; Outer radius =

12

feet.

The area formula requires the radius, which is half of the

20

-foot diameter. The outer radius includes the

10

-foot fountain radius plus the

2

-foot walkway.

2

Calculate the area of the concrete walkway.

(3.14 \times 12^2) - (3.14 \times 10^2) = 452.16 - 314 = 138.16

square feet.

The walkway forms a ring (annulus) around the fountain. Its area is the total outer area minus the inner fountain area.

3

Calculate the total cost of the concrete.

138.16 \text{ sq ft} \times \$5.00/\text{sq ft} = \$690.80

.

The total material cost is the square footage of the walkway multiplied by the price per square foot.

Key Concept

Calculating the area of a circular ring (annulus) and applying unit costs.

Question 47Question →

A military logistics facility is upgrading the security fence around a rectangular motor pool. The motor pool is $50$ yards long and $84$ feet wide. The new design requires leaving a $24$ -foot gap for a main vehicle gate and two separate $3$ -foot gaps for pedestrian access. The security fencing is purchased in pre-assembled panels that are $72$ inches long each. If each panel costs $\$120$ , what is the total cost of the fencing panels needed to enclose the motor pool?

Show answer & explanation

Answer: 8760

Answer

8760 dollars

The correct calculation requires converting all linear measurements to a common unit (feet), calculating the perimeter of the rectangle, subtracting the required gaps for the gates, dividing the remaining length by the panel length to find the total number of panels, and finally multiplying by the cost per panel.

Step-by-Step Solution

1

Convert yards and inches to feet.

Length is 150 feet; panel length is 6 feet.

To ensure consistency in perimeter and material calculations before determining the number of panels.

2

Calculate the full perimeter of the rectangular motor pool.

468 feet.

To find the total boundary length before accounting for the gaps left for gates.

3

Subtract the total width of the gate gaps from the full perimeter.

438 feet.

Because fencing panels are not installed where the vehicle and pedestrian gates are located.

4

Divide the required fencing length by the length of a single panel.

73 panels.

To find how many discrete 6-foot sections make up the 438-foot fence line.

5

Multiply the number of panels by the cost per panel.

$8,760.

To find the final total price of purchasing the required materials.

Key Concept

Calculating perimeter with unit conversions and subtracting gaps to find total material cost.

Estimated Time:2m 0s

Question 48Question →

A builder uses a 15-foot diagonal support beam to brace a vertical wall. If the base of the beam rests on the flat floor exactly 9 feet away from the bottom of the wall, at what height on the wall, in feet, does the top of the beam attach?

Show answer & explanation

Answer: 12

Answer

12

The physical scenario forms a right triangle where the hypotenuse (the beam) measures 15 feet and one leg (along the floor) measures 9 feet. By using the Pythagorean theorem (

a^2 + b^2 = c^2

), we can find the missing leg by solving

9^2 + b^2 = 15^2

. This simplifies to

81 + b^2 = 225

, which means

b^2 = 144

. Taking the square root of 144 gives a wall height of 12 feet.

Step-by-Step Solution

1

Identify the parts of the right triangle.

Hypotenuse

c = 15

feet, base leg

a = 9

feet, vertical leg

b

is unknown.

The beam forms the hypotenuse opposite the right angle made by the intersection of the vertical wall and flat floor.

2

Apply the Pythagorean theorem.

9^2 + b^2 = 15^2

The Pythagorean theorem (

a^2 + b^2 = c^2

) relates the lengths of the legs of a right triangle to its hypotenuse.

3

Square the known values.

81 + b^2 = 225

To solve for the unknown leg

b

, we must first evaluate the squared terms.

4

Isolate the unknown squared variable.

b^2 = 225 - 81 = 144

Subtracting 81 from both sides leaves

b^2

by itself.

5

Take the square root to find the height.

b = \sqrt{144} = 12

Taking the square root of

b^2

gives the actual length of the wall height.

Key Concept

Using the Pythagorean theorem to find the missing leg of a right triangle.

Question 49Question →

A local radio station broadcasts its signal in a circular area with a radius of $40$ miles. The station upgrades its transmitter, which increases the broadcasting radius to $50$ miles. How many additional square miles of coverage does the upgraded transmitter provide? (Use $\pi = 3.14$ )

Show answer & explanation

Answer: 2826

Answer

2826 square miles

The additional coverage is represented geometrically by the area of the ring (annulus) formed between the two circular boundaries. This is found by calculating both areas and subtracting the original area (

3.14 \times 40^2 = 5024

) from the new area (

3.14 \times 50^2 = 7850

), resulting in

2826

square miles of extra coverage.

Step-by-Step Solution

1

Calculate the original coverage area.

A_{original} = 3.14 \times 40^2 = 3.14 \times 1600 = 5024

square miles

We must determine the initial broadcast area before the upgrade.

2

Calculate the new coverage area.

A_{new} = 3.14 \times 50^2 = 3.14 \times 2500 = 7850

square miles

We need the total broadcast area provided by the new transmitter.

3

Calculate the additional coverage area by finding the difference.

7850 - 5024 = 2826

square miles

The additional area is the exact difference between the new area and the original area.

Key Concept

Calculating the difference between two circular areas (area of an annulus).

Alternative Method

Instead of calculating both areas separately, you can factor the expression for the difference in area:

\Delta A = \pi(R^2 - r^2) = 3.14 \times (50^2 - 40^2) = 3.14 \times (2500 - 1600) = 3.14 \times 900 = 2826

.

Estimated Time:1m 30s

Question 50Question →

A maintenance crew is installing a new rectangular cooling fluid reservoir for an industrial generator. The interior of the reservoir measures $4$ feet long, $3$ feet wide, and $18$ inches deep. If the crew fills the empty reservoir at a constant rate of $1.5$ cubic feet per minute, how many minutes will it take to fill the reservoir completely?

1.2

8

12

144

Show answer & explanation

Answer: 12

Answer

It will take 12 minutes to fill the reservoir completely.

The correct answer is

12

. To determine the fill time, the dimensions must first be unified. Since the flow rate is in cubic feet,

18

inches must be converted to

1.5

feet. The interior volume of the rectangular reservoir is then

4 \text{ ft} \times 3 \text{ ft} \times 1.5 \text{ ft} = 18 \text{ cubic feet}

. Dividing this total volume by the filling rate of

1.5

cubic feet per minute yields exactly

12

minutes.

Step-by-Step Solution

1

Convert the depth measurement from inches to feet.

18 \text{ inches} \div 12 = 1.5 \text{ feet}

.

All dimensions must be in the same unit (feet) because the flow rate is provided in cubic feet per minute.

2

Calculate the total volume of the rectangular reservoir.

4 \text{ ft} \times 3 \text{ ft} \times 1.5 \text{ ft} = 18 \text{ cubic feet}

.

The capacity of a rectangular container is found using the volume formula

V = l \times w \times h

.

3

Determine the time required to fill the reservoir.

18 \text{ cubic feet} \div 1.5 \text{ cubic feet per minute} = 12 \text{ minutes}

.

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

Key Concept

Calculating rectangular volume with unit conversions and applying rates.

Estimated Time:1m 30s

Question 51Question →

A rectangular fuel holding tank at an aviation depot is $15$ feet long and $10$ feet wide. The tank currently holds $900$ cubic feet of aviation fuel, which fills it to exactly half of its total capacity. What is the total interior height of the tank, in feet?

Show answer & explanation

Answer: 12

Answer

The total interior height of the tank is

12

feet.

By first finding the base area (

15 \times 10 = 150

sq ft), we can determine how high

900

cubic feet of fuel reaches by dividing the volume by the area (

900 \div 150 = 6

feet). Because the tank is half full, the

6

-foot depth is exactly half of the tank's total height, making the total height

12

feet.

Step-by-Step Solution

1

Calculate the area of the rectangular base of the tank.

15 \text{ ft} \times 10 \text{ ft} = 150 \text{ sq ft}

The volume of a rectangular prism is the product of its base area and height. Finding the base area is the first step to isolating the height.

2

Calculate the current depth of the aviation fuel.

900 \text{ cu ft} \div 150 \text{ sq ft} = 6 \text{ ft}

Dividing the current volume of the fuel by the base area of the tank yields the height (depth) of the liquid currently inside.

3

Calculate the total interior height of the tank.

6 \text{ ft} \times 2 = 12 \text{ ft}

The problem states the tank is exactly half full, meaning the total interior height is twice the depth of the current fuel level.

Key Concept

Calculating a missing dimension of a rectangular prism using volume and partial capacity

Question 52Question →

During a base expansion project, civil engineers are preparing a rectangular wooden form to cast a concrete foundation block. The interior of the form measures $12.5$ feet in length, $8$ feet in width, and $36$ inches in depth. If a mixer truck pumps concrete into the form at a constant rate of $20$ cubic feet per minute, how many minutes will it take to fill the form completely?

5

15

150

180

Show answer & explanation

Answer: $15$

Answer

It will take 15 minutes to completely fill the concrete form.

To find the time it takes to fill the form, first ensure all dimensions are in the same unit. The length and width are in feet, but the depth is in inches. Converting

36

inches to feet yields

3

feet. Next, calculate the total volume of the rectangular form using the formula

V = l \times w \times h

. Multiplying

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

gives a total volume of

300

cubic feet. Finally, divide this total volume by the pouring rate of

20

cubic feet per minute.

300 \div 20 = 15

minutes.

Step-by-Step Solution

1

Convert the depth from inches to feet to match the other dimensions and the unit of the flow rate.

36 \text{ inches} \div 12 \text{ inches/foot} = 3 \text{ feet}

Volume calculations require all linear dimensions to be in the same unit before multiplying.

2

Calculate the total volume of the rectangular form.

12.5 \text{ ft} \times 8 \text{ ft} \times 3 \text{ ft} = 300 \text{ cubic feet}

The capacity of a rectangular container is found by multiplying its length, width, and depth (

V = l \times w \times h

).

3

Calculate the time required to fill the form by dividing the total volume by the pumping rate.

300 \text{ cubic feet} \div 20 \text{ cubic feet/minute} = 15 \text{ minutes}

Time equals total volume divided by the rate of flow.

Key Concept

Calculating the volume of a rectangular prism (

V = l \times w \times h

) with proper unit conversions, and using rates to find total time.

Question 53Question →

A contractor is installing a decorative border around a newly built rectangular patio. The patio is $15$ feet long and $10$ feet wide. If the border material costs $\$2$ per foot, what is the total cost of the material needed?

\$50

\$100

\$150

\$300

Show answer & explanation

Answer: $\$100$

Answer

The total cost of the material needed is

\$100

.

The correct answer is found by first calculating the perimeter of the rectangle, which is

2 \times (15 + 10) = 50

feet. Then, this total length is multiplied by the price per foot (

50 \times \$2

) to get a final cost of

\$100

.

Step-by-Step Solution

1

Identify that the border goes around the outside of the patio, which means you need to find the perimeter.

Perimeter =

2 \times (\text{Length} + \text{Width})

Understanding the difference between the boundary (perimeter) and the inside space (area).

2

Calculate the total perimeter of the rectangular patio.

Perimeter =

2 \times (15 + 10) = 2 \times 25 = 50

feet.

This determines the total linear feet of border material required.

3

Multiply the total perimeter by the cost per foot to find the total cost.

50 \text{ feet} \times \$2\text{/foot} = \$100

.

Applying the unit rate to the total length gives the final financial cost.

Key Concept

Calculating the perimeter of a rectangle and applying a unit rate to find total cost.

Estimated Time:45s

Question 54Question →

A facility management team is filling a newly constructed rectangular water holding tank for the base fire department. The tank's interior measures $30$ feet long, $15$ feet wide, and $10$ feet deep. For safety regulations, the tank must only be filled to $80\%$ of its total capacity. If the water pump delivers $600$ cubic feet of water per hour, how many hours will it take to reach the required fill level?

Show answer & explanation

Answer: 6

Answer

6

The total volume of the tank is

30 \times 15 \times 10 = 4500

cubic feet. Because it must only be filled to

80\%

capacity, the required volume of water is

4500 \times 0.80 = 3600

cubic feet. At a pumping rate of

600

cubic feet per hour, the time needed to reach this level is

3600 \div 600 = 6

hours.

Step-by-Step Solution

1

Calculate the total volume of the holding tank.

Total Volume = 30 × 15 × 10 = 4500 cubic feet.

The volume of a rectangular prism is found using the formula V = length × width × height.

2

Calculate 80% of the total capacity.

Target Volume = 4500 × 0.80 = 3600 cubic feet.

Safety regulations require the tank to be filled to only 80% of its full capacity.

3

Determine the time required to pump the target volume.

Time = 3600 ÷ 600 = 6 hours.

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

Key Concept

Calculating rectangular volume, finding a percentage of a total, and applying flow rates to find time.

Alternative Method

Instead of finding the total volume first, you can find the target depth. 80% of the 10-foot depth is 8 feet. The target volume is then calculated using this adjusted depth: 30 × 15 × 8 = 3600 cubic feet. Divide this by 600 to get 6 hours.

Estimated Time:1m 30s

Question 55Question →

A volunteer group is installing a wooden fence around a rectangular community garden. The garden measures $30$ feet in length and $18$ feet in width. How many feet of fencing are needed to completely enclose the garden?

Show answer & explanation

Answer: 96

Answer

96

The amount of fencing needed to completely enclose a rectangular space is equal to the perimeter of that space. The formula for the perimeter of a rectangle is

P = 2 \times (L + W)

. By substituting the given length (

30

feet) and width (

18

feet), the perimeter is

2 \times (30 + 18) = 2 \times 48 = 96

feet.

Step-by-Step Solution

1

Identify the formula for the perimeter of a rectangle.

P = 2 \times (\text{Length} + \text{Width})

The total fencing needed to completely enclose a rectangular area is equal to its perimeter.

2

Substitute the given dimensions into the formula.

P = 2 \times (30 + 18) = 2 \times 48 = 96

This calculates the total linear feet of the boundary for a garden with a length of 30 feet and a width of 18 feet.

Key Concept

The total length of fencing required to enclose a rectangular area is equal to its perimeter, calculated as

2 \times (\text{length} + \text{width})

.

Question 56Question →

An engineer battalion is excavating a rectangular subterranean bunker. The excavated space must be $120$ feet long, $30$ feet wide, and $15$ feet deep. If the earth-moving equipment removes soil at a rate of $200$ cubic yards per hour, how many hours will it take to complete the excavation?

270

90

18

10

Show answer & explanation

Answer: 10

Answer

It will take 10 hours to complete the excavation.

The correct answer is

10

. To find the time required, we must determine the total volume of the bunker in cubic yards, since the removal rate is given in cubic yards per hour. First, convert the dimensions from feet to yards by dividing by

3

: the length is

40

yards, the width is

10

yards, and the depth is

5

yards. The volume is

40 \times 10 \times 5 = 2,000

cubic yards. Dividing the total volume of

2,000

cubic yards by the rate of

200

cubic yards per hour yields

10

hours.

Step-by-Step Solution

1

Convert the bunker dimensions from feet to yards.

Length =

40

yards, Width =

10

yards, Depth =

5

yards.

The removal rate is given in cubic yards, so calculating the volume directly in cubic yards simplifies the problem.

2

Calculate the total volume of the bunker.

40 \text{ yd} \times 10 \text{ yd} \times 5 \text{ yd} = 2,000

cubic yards.

The total volume represents the amount of soil that needs to be removed.

3

Calculate the time required by dividing the volume by the removal rate.

2,000 \text{ yd}^3 \div 200 \text{ yd}^3/\text{hr} = 10

hours.

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

Key Concept

Converting linear dimensions to calculate volume in a specific unit, and using rates to find time.

Alternative Method

Instead of converting the linear dimensions first, you can calculate the volume in cubic feet (

120 \times 30 \times 15 = 54,000 \text{ ft}^3

), convert the removal rate to cubic feet per hour (

200 \text{ yd}^3/\text{hr} \times 27 \text{ ft}^3/\text{yd}^3 = 5,400 \text{ ft}^3/\text{hr}

), and then divide:

54,000 \div 5,400 = 10

hours.

Estimated Time:1m 30s

Question 57Question →

A school is installing a new fence around a square play area. Each side of the play area measures $5$ yards. If the fencing material costs $\$0.50$ per foot, what is the total cost of the fencing required?

\$10.00

\$30.00

\$112.50

\$300.00

Show answer & explanation

Answer: $\$30.00$

Answer

\$30.00

The correct answer is found by first ensuring the units match. Since the cost is provided per foot, convert the

5

-yard sides into feet (

5 \times 3 = 15

feet). Next, find the perimeter of the square play area (

15 \times 4 = 60

feet). Finally, multiply the

60

-foot perimeter by the

\$0.50

per foot cost to determine a total fencing cost of

\$30.00

.

Step-by-Step Solution

1

Convert the side length from yards to feet.

5 \text{ yards} \times 3 = 15 \text{ feet}

The cost of fencing is given per foot, so all geometric measurements must be converted to feet before calculating the total cost.

2

Calculate the perimeter of the square play area.

4 \times 15 \text{ feet} = 60 \text{ feet}

The fence goes around the outer boundary, which is the perimeter of the square.

3

Calculate the total cost by multiplying the perimeter by the cost per foot.

60 \text{ feet} \times \$0.50/\text{foot} = \$30.00

To find the total cost, multiply the total linear feet of fencing needed by the given price per foot.

Key Concept

Calculating perimeter of a square and applying unit conversions to determine total cost.

Question 58Question →

A rancher is building a fence around a rectangular grazing pasture. The pasture is $140$ feet long and $70$ feet wide. If the rancher needs to leave a $10$ -foot wide opening for a gate, how many total feet of fencing material will be required?

Show answer & explanation

Answer: 410

Answer

410 feet

To find the total amount of fencing required, you first determine the perimeter of the rectangle, which is

140 + 140 + 70 + 70 = 420

feet. Because there is a

10

-foot gate that does not require fencing, you subtract

10

from the total perimeter, leaving

410

feet.

Step-by-Step Solution

1

Calculate the total perimeter of the rectangular pasture.

2 \times (140 + 70) = 420

feet.

The fence runs along the entire outer boundary of the pasture, which requires finding the perimeter of the rectangle.

2

Subtract the gate opening from the total perimeter.

420 - 10 = 410

feet.

The

10

-foot opening for the gate does not use fencing material, so it must be subtracted from the total.

Key Concept

Calculating perimeter of a rectangle and adjusting for gaps or openings.

Question 59Question →

Service members are constructing a rectangular wooden planter box for a community project. The interior dimensions of the planter are $16$ feet in length and $4$ feet in width. A delivery of $128$ cubic feet of potting soil fills the planter perfectly to a level exactly $6$ inches below its top edge. What is the total interior height of the planter, in feet?

2.0

2.5

2.6

8.0

Show answer & explanation

Answer: $2.5$

Answer

The total interior height of the planter is

2.5

feet.

To find the total height, first determine the height of the soil. Since Volume = Length × Width × Height, the soil height is

128 \div (16 \times 4) = 2

feet. The problem states there is a

6

-inch gap at the top. Converting

6

inches to feet gives

0.5

feet. Adding the soil height and the gap gives a total interior height of

2 + 0.5 = 2.5

feet.

Step-by-Step Solution

1

Calculate the area of the planter's rectangular base.

16 \times 4 = 64

square feet.

The base area is required to determine how deep the potting soil will sit.

2

Calculate the depth of the potting soil.

128 \div 64 = 2

feet.

Dividing the total volume of soil by the base area yields the height (or depth) of the soil layer.

3

Convert the remaining empty height from inches to feet.

6 \text{ inches} = 0.5 \text{ feet}

.

All measurements must be in the same unit (feet) before they can be added together.

4

Add the soil depth to the empty space depth.

2 + 0.5 = 2.5

feet.

The total interior height is the sum of the filled space and the empty space.

Key Concept

Solving for a missing dimension in a rectangular volume and ensuring measurement units are consistent.

Question 60Question →

A city construction crew is installing a new cylindrical concrete drainage pipe. The interior of the pipe has a diameter of $10\text{ feet}$ and a length of $100\text{ feet}$ . Using $3.14$ for $\pi$ , what is the maximum volume of water the pipe can hold when completely full, in cubic feet?

Show answer & explanation

Answer: 7850

Answer

The pipe can hold a maximum volume of 7,850 cubic feet.

The volume of a cylinder is found by multiplying the area of its circular base by its height or length (

V = \pi r^2 h

). First, convert the given diameter of

10\text{ feet}

to a radius by dividing by

2

, yielding a radius of

5\text{ feet}

. Next, square the radius to get

25

. Multiply this by the length (

100\text{ feet}

) and the approximate value of

\pi

(

3.14

).

3.14 \times 25 \times 100 = 7,850\text{ cubic feet}

.

Step-by-Step Solution

1

Find the radius of the pipe.

r = 10 \div 2 = 5\text{ feet}

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

2

Apply the volume formula for a cylinder.

V = \pi \times r^2 \times h

To find the capacity of the cylindrical pipe, we must calculate its three-dimensional space, which is volume.

3

Substitute the known values into the formula.

V = 3.14 \times 5^2 \times 100

We use the given values: radius

5\text{ feet}

, height (or length)

100\text{ feet}

, and

\pi \approx 3.14

.

4

Calculate the final volume.

V = 3.14 \times 25 \times 100 = 3.14 \times 2500 = 7850

Squaring the radius gives

25

. Multiplying by

100

and

3.14

gives the total volume in cubic feet.

Key Concept

Calculating the volume of a cylinder using the formula

V = \pi r^2 h

, ensuring the given diameter is converted to a radius before performing calculations.

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