Question

Question
1
Question
2
A)
How do you derive the formula for the area of a sector?
(1 point)
Divide the measurement of the sector’s angle by
, then multiply the quotient by
times the radiussquared.
180
°
π
Divide the measurement of the sector’s angle by
, then multiply the quotient by
times the radius.
360
°
π
Divide the measurement of the sector’s angle by
, then divide the quotient by
times the radiussquared.
360
°
π
Divide the measurement of the sector’s angle by
, then multiply the quotient by
times the radiussquared.
360
°
π
A)
What is the area of the sector of a circle with
and a radius of 4 feet? Include pi in yourcalculation.
(1 point)
θ = 75°
A
=
2.168
ft.
2
A
=
50.265
ft.
2
A
=
10.472
ft.
2
A
=
39.794
ft.
2
Question
3
Question
4
Question
5
A)
Using the formula for the area of a sector, solve for the area of a sector with an angle of 135° and aradius of 13
(1 point) in.
A
=
398.2
in.
2
A
=
199.1
in.
2
A
=
530.9
in.
2
A
=
30.6
in.
2
A)
The radius of a circle is 6
Find the area of a sector with an angle of 120°.
(1 point) in.
A
=
12
π
in.
2
A
=
9
π
in.
2
A
=
36
π
in.
2
A
=
2
π
in.
2
A)
There is a sprinkler in Amut’s backyard that can shoot water a distance of 15 feet from the sprinklerhead. It rotates on the center point at an angle of 180° before returning to its starting position. Whatarea of the backyard, in feet, can the sprinkler spray?
(1 point)
A
=
112.5
π
ft.
2
A
=
56.25
π
ft.
2
A
=
7.5
π
ft.
2
A
=
225
π
ft.
2

Answer 1

Here are the answers with brief work:

1. How to derive the sector area formula:
- Divide the sector angle by 360°, then multiply that quotient by πr^2.

2. Area for θ = 75°, r = 4 ft:
- A = (75/360)·π·4^2 = (5/24)·16π = (10/3)π ≈ 10.472 ft^2 → 10.472 ft^2

3. Area for θ = 135°, r = 13 in:
- A = (135/360)·π·13^2 = (3/8)·169π = 63.375π ≈ 199.1 in^2 → 199.1 in^2

4. Radius 6 in, θ = 120°:
- A = (120/360)·π·6^2 = (1/3)·36π = 12π in^2 → 12π in^2

5. Sprinkler r = 15 ft, θ = 180°:
- A = (180/360)·π·15^2 = 1/2·225π = 112.5π ft^2 → 112.5π ft^2