Sure, let's solve this problem step by step.
Given:
- The diameter of the tire is 48 inches.
- The tire rotates through an angle of 94 degrees.
Step 1: Calculate the circumference of the tire.
The circumference \( C \) of a circle is given by the formula:
[tex]\[
C = \pi \times \text{diameter}
\][/tex]
Substituting the given diameter:
[tex]\[
C = \pi \times 48
\][/tex]
So, the circumference of the tire is:
[tex]\[
C \approx 150.796 \, \text{inches}
\][/tex]
Step 2: Determine the fraction of the circle that 94 degrees represents.
A full circle is 360 degrees. To find the fraction of the circle that 94 degrees represents, we divide 94 by 360:
[tex]\[
\text{Fraction of the circle} = \frac{94}{360}
\][/tex]
Calculating this fraction:
[tex]\[
\text{Fraction of the circle} \approx 0.2611
\][/tex]
Step 3: Calculate the distance traveled by the rock.
The distance traveled is the fraction of the circumference. Therefore, we multiply the circumference by the fraction of the circle:
[tex]\[
\text{Distance traveled} = C \times \text{Fraction of the circle}
\][/tex]
Substitute the known values:
[tex]\[
\text{Distance traveled} = 150.796 \times 0.2611
\][/tex]
Calculating the distance:
[tex]\[
\text{Distance traveled} \approx 39.375 \, \text{inches}
\][/tex]
So, the rock travels approximately 39.375 inches when the tire rotates \( 94^\circ \).
Given the multiple-choice options, none directly match the calculated distance of 39.375 inches in its simplest form, so the correct answer is not provided directly but derived from the calculations above.
