The tip of a pendulum swings through a central angle measuring \frac{\pi}{3} radians, trac… — ACT

The tip of a pendulum swings through a central angle measuring $\frac{\pi}{3}$ radians, tracing an arc length of $4\pi$ centimeters. What is the area, in square centimeters, of the sector swept out by the pendulum?

A
$12\pi$
$24\pi$ Answer
C
$36\pi$
D
$48\pi$
E
$144\pi$

Answer

24\pi

square centimeters

To find the area of the sector, first determine the radius of the pendulum's circular path using the arc length formula

s = r\theta

. Substituting

s = 4\pi

and

\theta = \frac{\pi}{3}

gives

r = 12

cm. Then, substitute the radius and central angle into the sector area formula

A = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2\left(\frac{\pi}{3}\right) = 24\pi

square centimeters.

Step-by-Step Solution

Calculate the radius of the circle using the arc length formula in radians.

Using

s = r\theta

with

s = 4\pi

and

\theta = \frac{\pi}{3}

gives

4\pi = r\left(\frac{\pi}{3}\right) \implies r = 12

cm.

The radius is required to calculate the sector area.

Calculate the area of the sector using the radian sector area formula.

A = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(144)\left(\frac{\pi}{3}\right) = 24\pi

square centimeters.

Multiplying half the squared radius by the radian measure of the central angle yields the sector area.

Key Concept

Arc Length and Sector Area in Radians

Estimated Time:1m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic