A searchlight beam sweeps horizontally across a central angle of 72^\circ. If the maximum … — Armed Services Vocational Aptitude Battery (ASVAB)

A searchlight beam sweeps horizontally across a central angle of $72^\circ$ . If the maximum effective range of the searchlight is $20 \text{ meters}$ , what is the length of the arc formed by the outer edge of the beam? (Leave your answer in terms of $\pi$ ).

A
$4\pi \text{ meters}$
$8\pi \text{ meters}$ Answer
C
$40\pi \text{ meters}$
D
$80\pi \text{ meters}$

Answer

8\pi \text{ meters}

The length of an arc is calculated by multiplying the full circumference of the circle (

2\pi r

) by the fraction of the circle represented by the central angle (

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

). In this scenario, the full circumference is

2\pi(20) = 40\pi \text{ meters}

. The fraction representing the

72^\circ

sweep is

\frac{72}{360}

, which simplifies to

\frac{1}{5}

. Multiplying the full circumference by this fraction gives

\frac{1}{5} \times 40\pi = 8\pi \text{ meters}

Step-by-Step Solution

Identify the formula for the arc length of a circle.

\text{Arc Length} = (\frac{\theta}{360^\circ}) \times 2\pi r

The arc length represents a specific fraction of the entire circumference based on the central angle.

Determine the fraction of the circle covered by the searchlight beam.

\frac{72^\circ}{360^\circ} = \frac{1}{5}

Simplifying the angle fraction makes the subsequent multiplication easier.

Calculate the full circumference of the circle using the given radius (range).

C = 2\pi(20) = 40\pi \text{ meters}

The full circumference is necessary to find the length of the fractional arc.

Multiply the full circumference by the angle fraction.

\frac{1}{5} \times 40\pi = 8\pi \text{ meters}

This calculation applies the proportional fraction to the total boundary of the circle, yielding the arc length.

Key Concept

Arc Length of a Circle

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