A tire on a car has an outside diameter of 37.64 in. Through what angle (in radians) does the tire turn while traveling 1 mile? The tire turns \( \square \) radians. (Round to the nearest hundredth as needed.)

To find the angle in radians through which the tire turns while traveling 1 mile, we first need to calculate the circumference of the tire. The formula for the circumference \( C \) of a circle is given by \( C = \pi \times d \), where \( d \) is the diameter. Given that the diameter of the tire is 37.64 inches, the circumference \( C \) will be: \[ C = \pi \times 37.64 \approx 118.15 \text{ inches} \] Now, we need to convert the distance traveled (1 mile) into inches. There are 63,360 inches in 1 mile, so: \[ \text{Distance} = 63,360 \text{ inches} \] Next, to find out how many times the tire turns while traveling this distance, we divide the total distance by the circumference of the tire: \[ \text{Number of turns} = \frac{63,360 \text{ inches}}{118.15 \text{ inches}} \approx 537.23 \] Since each complete turn of the tire corresponds to an angle of \( 2\pi \) radians, we can find the total angle in radians by multiplying the number of turns by \( 2\pi \): \[ \text{Angle in radians} = 537.23 \times 2\pi \approx 3376.12 \text{ radians} \] Therefore, the tire turns approximately \( 3376.12 \) radians while traveling 1 mile.