The tip of a mechanical pendulum swings along a circular arc, sweeping out a sector of a c… — ACT

The tip of a mechanical pendulum swings along a circular arc, sweeping out a sector of a circle. The arc length traveled by the tip of the pendulum is $8\pi$ inches, and the area of the circular sector swept out is $48\pi$ square inches. What is the total perimeter, in inches, of this circular sector?

A
$12 + 8\pi$
$24 + 8\pi$ Answer
C
$48 + 8\pi$
D
$8\pi$
E
$24 + 4\pi$

Answer

The total perimeter of the circular sector is

24 + 8\pi

inches.

The area of a circular sector with radius

r

and arc length

s

is given by

A = \frac{1}{2} r s

. Substituting the given values

A = 48\pi

and

s = 8\pi

gives

48\pi = \frac{1}{2} r (8\pi)

, which simplifies to

48\pi = 4\pi r

, so

r = 12

inches. The total perimeter of the sector consists of the arc length plus two straight radii:

P = 2r + s = 2(12) + 8\pi = 24 + 8\pi

inches.

Step-by-Step Solution

Relate sector area, radius, and arc length

Use the formula

\text{Area} = \frac{1}{2} r s

, where

s

is the arc length and

r

is the radius.

This direct relationship allows solving for the radius without needing to calculate the central angle explicitly.

Solve for the radius

r

48\pi = \frac{1}{2} \cdot r \cdot (8\pi) \implies 48\pi = 4\pi r \implies r = 12

inches.

Dividing both sides by

4\pi

determines the length of the pendulum arm (the radius).

Calculate the total sector perimeter

\text{Perimeter} = 2r + s = 2(12) + 8\pi = 24 + 8\pi$ inches.

The boundary of a circular sector consists of two straight radii and the curved arc.

Key Concept

Relationship between Sector Area, Arc Length, Radius, and Sector Perimeter

Estimated Time:1m 30s

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