A circular radar display at an air traffic control tower has a radius of 12 inches. A wedg… — ACT

A circular radar display at an air traffic control tower has a radius of $12$ inches. A wedge-shaped tracking zone on the display is bounded by two radii and an outer arc length of $5\pi$ inches. What is the area, in square inches, of this tracking zone?

A
$15\pi$
$30\pi$ Answer
C
$60\pi$
D
$120\pi$
E
$144\pi$

Answer

The area of the tracking zone is

30\pi

square inches.

The total area of the circle is

\pi(12)^2 = 144\pi

square inches, and the total circumference is

2\pi(12) = 24\pi

inches. The tracking sector accounts for a fraction of

\frac{5\pi}{24\pi} = \frac{5}{24}

of the full circle. Multiplying this fraction by the total area yields

\frac{5}{24} \times 144\pi = 30\pi

square inches.

Step-by-Step Solution

Calculate the total circumference and total area of the circular display

Circumference

C = 2\pi r = 2\pi(12) = 24\pi

inches, and Total Area

A = \pi r^2 = \pi(12)^2 = 144\pi

square inches.

These total values establish the scale needed to find the fractional sector area.

Find the fraction of the circle represented by the outer arc length

Fraction

= \frac{\text{Arc Length}}{\text{Circumference}} = \frac{5\pi}{24\pi} = \frac{5}{24}

The ratio of an arc length to the total circumference equals the ratio of the sector area to the total area.

Multiply the fraction by the total area of the circle

\text{Sector Area} = \frac{5}{24} \times 144\pi = 5 \times 6\pi = 30\pi$ square inches.

Applying the arc length fraction to the total area yields the exact sector area.

Key Concept

Sector Area and Arc Length Proportions

Estimated Time:1m 0s

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