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Question 1

A)How do you derive the formula for the area of a sector?(1 point)
Responses

Divide the measurement of the sector’s angle by 180°, then multiply the quotient by π times the radius squared.
Divide the measurement of the sector’s angle by 180 degrees , then multiply the quotient by pi times the radius squared.

Divide the measurement of the sector’s angle by 360°, then multiply the quotient by π times the radius.
Divide the measurement of the sector’s angle by 360 degrees , then multiply the quotient by pi times the radius.

Divide the measurement of the sector’s angle by 360°, then multiply the quotient by π times the radius squared.
Divide the measurement of the sector’s angle by 360 degrees , then multiply the quotient by pi times the radius squared.

Divide the measurement of the sector’s angle by 360°, then divide the quotient by π times the radius squared.
Divide the measurement of the sector’s angle by 360 degrees , then divide the quotient by pi times the radius squared.
Question 2
A)What is the area of the sector of a circle with θ=75° and a radius of 4 feet? Include pi in your calculation.(1 point)
Responses

A=2.168 ft.2
upper A equals 2.168 ft. squared

A=50.265 ft.2
upper A equals 20.265 ft. squared

A=39.794 ft.2
upper A equals 39.794 ft. squared

A=10.472 ft.2
upper A equals 10.472 ft. squared
Question 3
A)Using the formula for the area of a sector, solve for the area of a sector with an angle of 135° and a radius of 13 in.(1 point)
Responses

A=30.6 in.2
upper A equals 30.6 in. squared

A=199.1 in.2
upper A equals 199.1 in. squared

A=398.2 in.2
upper A equals 398.2 in. squared

A=530.9 in.2
upper A equals 530.9 in. squared
Question 4
A)The radius of a circle is 6 in. Find the area of a sector with an angle of 120°.(1 point)
Responses

A=12π in.2
upper A equals 12 pi in. squared

A=2π in.2
upper A equals 2 pi in. squared

A=36π in.2
upper A equals 36 pi in. squared

A=9π in.2
upper A equals 9 pi in. squared
Question 5
A)There is a sprinkler in Amut’s backyard that can shoot water a distance of 15 feet from the sprinkler head. It rotates on the center point at an angle of 180° before returning to its starting position. What area of the backyard, in feet, can the sprinkler spray?(1 point)
Responses

A=56.25π ft.2
upper A equals 56.25 pi ft. squared

A=112.5π ft.2
upper A equals 112.5 pi ft. squared

A=7.5π ft.2

1 answer

Let's go through each question one by one to derive the formulas and calculate the areas.

Question 1

The formula for the area of a sector is given by:

\[ A = \frac{\theta}{360°} \times \pi r^2 \]

The correct response is: Divide the measurement of the sector’s angle by 360°, then multiply the quotient by π times the radius squared.

Question 2

Given:

  • Angle \( \theta = 75° \)
  • Radius \( r = 4 \) ft

Using the formula for the area of a sector:

\[ A = \frac{75}{360} \times \pi \times (4)^2 \] \[ A = \frac{75}{360} \times \pi \times 16 \] \[ A = \frac{75 \times 16}{360} \times \pi = \frac{1200}{360} \times \pi = \frac{10}{3} \pi \approx 10.472 \text{ ft}^2 \]

The correct response is: A=10.472 ft.²

Question 3

Given:

  • Angle \( \theta = 135° \)
  • Radius \( r = 13 \) in

Using the formula for the area of a sector:

\[ A = \frac{135}{360} \times \pi \times (13)^2 \] \[ A = \frac{135}{360} \times \pi \times 169 \] \[ A = \frac{135 \times 169}{360} \times \pi = \frac{22815}{360} \times \pi \approx 63.375 \pi \approx 199.1 \text{ in}^2 \]

The correct response is: A=199.1 in.²

Question 4

Given:

  • Radius \( r = 6 \) in
  • Angle \( \theta = 120° \)

Using the formula for the area of a sector:

\[ A = \frac{120}{360} \times \pi \times (6)^2 \] \[ A = \frac{120}{360} \times \pi \times 36 \] \[ A = \frac{120 \times 36}{360} \times \pi = \frac{4320}{360} \times \pi = 12 \pi \text{ in}^2 \]

The correct response is: A=12π in.²

Question 5

Given:

  • Radius \( r = 15 \) ft
  • Angle \( \theta = 180° \)

Using the formula for the area of a sector:

\[ A = \frac{180}{360} \times \pi \times (15)^2 \] \[ A = \frac{1}{2} \times \pi \times 225 \] \[ A = \frac{225}{2} \times \pi = 112.5\pi \text{ ft}^2 \]

The correct response is: A=112.5π ft.²

Summary

  1. Divide by 360° for area of sector.
  2. A=10.472 ft.²
  3. A=199.1 in.²
  4. A=12π in.²
  5. A=112.5π ft.²