A rotary lawn sprinkler sweeps out a sector-shaped region of a yard with a central angle o… — ACT

A rotary lawn sprinkler sweeps out a sector-shaped region of a yard with a central angle of $150^\circ$ . If the area of the irrigated sector is $60\pi$ square feet, what is the perimeter, in feet, of the irrigated lawn sector?

A
$10\pi$
B
$10\pi + 12$
$10\pi + 24$ Answer
D
$5\pi + 24$
E
$20\pi + 24$

Answer

The perimeter of the irrigated lawn sector is

10\pi + 24

feet.

The area of a sector with central angle

150^\circ

\frac{150^\circ}{360^\circ} \times \pi r^2 = \frac{5}{12}\pi r^2

. Setting this equal to

60\pi

yields

r^2 = 144

, so the radius

r = 12

feet. The arc length is

\frac{5}{12} \times 2\pi(12) = 10\pi

feet. The perimeter of a sector is the arc length plus two radii (

2r

), giving

10\pi + 2(12) = 10\pi + 24

feet.

Step-by-Step Solution

Find the radius of the circle using the sector area formula.

r = 12

feet

The formula for sector area is

\text{Area} = \frac{\theta}{360^\circ} \cdot \pi r^2

. Substituting

\theta = 150^\circ

and

\text{Area} = 60\pi

gives

\frac{150}{360} \cdot \pi r^2 = 60\pi

, which simplifies to

\frac{5}{12} r^2 = 60

, so

r^2 = 144

and

r = 12

Calculate the arc length of the sector.

Arc length

= 10\pi

feet

The formula for arc length is

s = \frac{\theta}{360^\circ} \cdot 2\pi r

. Substituting

\theta = 150^\circ

and

r = 12

yields

s = \frac{5}{12} \cdot 2\pi (12) = 10\pi

Calculate the total perimeter of the sector.

Perimeter

= 10\pi + 24

feet

The perimeter of a sector consists of the curved arc length plus two straight radial edges:

\text{Perimeter} = s + 2r = 10\pi + 2(12) = 10\pi + 24

Key Concept

Perimeter of a Circle Sector

Estimated Time:1m 30s

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