A lighthouse beacon sweeps across a circular region of the ocean. The illuminated area for… — ACT

A lighthouse beacon sweeps across a circular region of the ocean. The illuminated area forms a circular sector with a radius of $15$ nautical miles and an area of $75\pi$ square nautical miles. What is the total perimeter, in nautical miles, of this illuminated sector?

A
$10\pi$
B
$10\pi + 15$
$10\pi + 30$ Answer
D
$20\pi + 30$
E
$5\pi + 30$

Answer

10\pi + 30

nautical miles

The full circle has an area of

\pi(15)^2 = 225\pi

. The illuminated sector's area of

75\pi

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

of the total circle. Thus, its arc length is

\frac{1}{3}

of the total circumference

2\pi(15) = 30\pi

, giving an arc length of

10\pi

. Adding the two bounding radii of length

15

each yields a total sector perimeter of

10\pi + 30

nautical miles.

Step-by-Step Solution

Calculate the total area of the full circle.

Total Area =

\pi r^2 = \pi (15)^2 = 225\pi

square nautical miles.

Knowing the full circle area allows us to find what fraction of the circle the sector occupies.

Determine the fractional proportion of the sector relative to the whole circle.

Fraction =

\frac{\text{Sector Area}}{\text{Total Area}} = \frac{75\pi}{225\pi} = \frac{1}{3}

The sector represents

\frac{1}{3}

of both the total area and the total circumference of the circle.

Calculate the arc length of the sector.

Arc length =

\frac{1}{3} \times 2\pi r = \frac{1}{3} \times 30\pi = 10\pi

nautical miles.

The curved boundary of the sector is

\frac{1}{3}

of the circle's full circumference.

Calculate the total perimeter of the sector by adding the arc length and the two straight radii.

Perimeter = Arc length

+ 2r = 10\pi + 15 + 15 = 10\pi + 30

nautical miles.

The perimeter of a circular sector consists of its curved outer arc plus its two straight bounding radii.

Key Concept

Perimeter of a Circular Sector

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