Circular Area and Circumference Applications Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB)

Soru 1Soru →

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

Cevabı ve açıklamayı göster

Cevap: $\$400\pi$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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.

Tahmini Süre:2m 0s

Soru 2Soru →

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 $1 \text{ mile} = 5,280 \text{ feet}$ .)

Cevabı ve açıklamayı göster

Cevap: 1080

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:2m 0s

Soru 3Soru →

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

Cevabı ve açıklamayı göster

Cevap: $28.26$ square meters

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Calculating the area of a circle given its diameter.

Soru 4Soru →

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

Cevabı ve açıklamayı göster

Cevap: 50.24

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Calculating the area of a circle given its diameter.

Soru 5Soru →

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

Cevabı ve açıklamayı göster

Cevap: $125.6$ feet

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:45s

Soru 6Soru →

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 $\pi = 3.14$ .

Cevabı ve açıklamayı göster

Cevap: 31.4

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Circumference of a circle

Tahmini Süre:45s

Soru 7Soru →

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

Cevabı ve açıklamayı göster

Cevap: $113.04$

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

.

Soru 8Soru →

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 $\pi = 3.14$ .

Cevabı ve açıklamayı göster

Cevap: 3140

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Circumference and Wheel Revolutions

Soru 9Soru →

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

Cevabı ve açıklamayı göster

Cevap: $314$ feet

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Distance from Rotations (Circumference)

Soru 10Soru →

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 $\pi = 3.14$ )

Cevabı ve açıklamayı göster

Cevap: 62.8

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Calculating and subtracting circular areas

Soru 11Soru →

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

Cevabı ve açıklamayı göster

Cevap: \$690.80

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 12Soru →

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 $\pi = 3.14$ )

Cevabı ve açıklamayı göster

Cevap: 2826

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Alternatif Yöntem

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

.

Tahmini Süre:1m 30s

Circular Area and Circumference Applications

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem