Caroline has a rock stuck in her Jeep's tire. If the tire has a 48-inch diameter, how far does the rock travel in [tex]$94^{\circ}$[/tex] of rotation?

A. [tex]\frac{188 \pi}{15}[/tex]
B. [tex]\frac{47 \pi}{90}[/tex]
C. [tex]24 \pi[/tex]
D. [tex]47 \pi[/tex]



Answer :

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.