Questions: A giant wheel has a radius of 14 ft. To the nearest tenth of a foot, what distance does the giant wheel cover when it rotates through an angle of 96°?

A giant wheel has a radius of 14 ft. To the nearest tenth of a foot, what distance does the giant wheel cover when it rotates through an angle of 96°?

Transcript text: A giant wheel has a radius of 14 ft. To the nearest tenth of a foot, what distance does the giant wheel cover when it rotates through an angle of $96^{\circ}$?

Solution

Solution Steps

To find the distance covered by the wheel when it rotates through a given angle, we can use the formula for the arc length of a circle. The arc length $ L $ is given by the formula $ L = \theta \times r $, where $ \theta $ is the angle in radians and $ r $ is the radius of the circle. First, convert the angle from degrees to radians, then calculate the arc length using the radius provided.

Step 1: Convert Angle from Degrees to Radians

To find the arc length, we first need to convert the angle from degrees to radians. The conversion formula is:

\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]

Given that $\theta_{\text{degrees}} = 96$, we have:

\[ \theta_{\text{radians}} = 96 \times \frac{\pi}{180} \approx 1.6755 \]

Step 2: Calculate the Arc Length

The arc length $L$ of a circle is given by the formula:

\[ L = \theta \times r \]

where $\theta$ is the angle in radians and $r$ is the radius of the circle. Substituting the given values, $\theta = 1.6755$ and $r = 14$ ft, we get:

\[ L = 1.6755 \times 14 \approx 23.4572 \]

Step 3: Round the Arc Length

Round the arc length to the nearest tenth: