A windshield wiper of length 15\text{ inches} sweeps through a central angle of 120^\circ … — ACT

A windshield wiper of length $15\text{ inches}$ sweeps through a central angle of $120^\circ$ across a windshield. What is the area, in square inches, of the region swept by the wiper?

A
$10\pi$
$75\pi$ Answer
C
$112.5\pi$
D
$150\pi$
E
$225\pi$

Answer

75\pi

square inches

The area of the region swept by the wiper is the area of a circle sector with a radius of

15\text{ inches}

and a central angle of

120^\circ

. The formula for the area of a sector is

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

. Substituting

15

for the radius and

120

for the angle gives

A = \frac{120}{360} \times \pi (15)^2 = \frac{1}{3} \times 225\pi = 75\pi

square inches.

Step-by-Step Solution

Determine the formula for the area of a sector of a circle.

A = \frac{\theta}{360^\circ} \times \pi r^2

The area of a sector is a fraction of the total area of the circle, where the fraction is determined by the central angle

\theta

divided by the total degrees in a circle (

360^\circ

Substitute the given values into the sector area formula, using a radius of

15

and a central angle of

120^\circ

A = \frac{120^\circ}{360^\circ} \times \pi (15)^2

The windshield wiper length represents the radius

r = 15\text{ inches}

, and the sweep angle represents the central angle

\theta = 120^\circ

Simplify the expression to find the final area.

A = \frac{1}{3} \times 225\pi = 75\pi

Reducing the fraction

\frac{120}{360}

\frac{1}{3}

and squaring

15

to get

225

yields the area of

75\pi

square inches.

Key Concept

The area of a circle sector is found by multiplying the total circle area,

\pi r^2

, by the ratio of the central angle to the total degree measure of a circle,

\frac{\theta}{360^\circ}

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