A dog is tied to a post in the center of a flat, grassy yard with a leash that is 12\text{… — ACT

A dog is tied to a post in the center of a flat, grassy yard with a leash that is $12\text{ feet}$ long. If the dog walks along an arc formed by a central angle of $\frac{\pi}{3}$ radians, what is the length, in feet, of the path the dog travels?

A
$2\pi$
$4\pi$ Answer
C
$8\pi$
D
$12\pi$
E
$24\pi$

Answer

The correct option is the one stating

4\pi

The length of the path traveled by the dog is the arc length of a circle with a radius of

12

feet and a central angle of

\frac{\pi}{3}

radians. Using the radian arc length formula,

s = r\theta

, substituting

r = 12

and

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

yields

s = 12 \times \frac{\pi}{3} = 4\pi

feet.

Step-by-Step Solution

Identify the given values from the problem statement.

The radius of the circular path is

r = 12

feet, and the central angle is

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

radians.

These parameters are required to calculate the arc length of the path.

Recall the formula for the arc length of a circle when the angle is in radians.

The formula is

s = r\theta

Since the angle is given in radians, the arc length is directly the product of the radius and the angle.

Substitute the values into the formula and calculate the result.

s = 12 \times \frac{\pi}{3} = 4\pi

Multiplying the radius by the angle in radians gives the length of the path traveled.

Key Concept

Calculating the arc length of a circle when the central angle is measured in radians using the formula

s = r\theta

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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