Question
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.)
Ask by Ball Munoz.
Jan 29,2025 00:29
UpStudy AI Solution
Tutor-Verified Answer
Answer
The tire turns approximately 3366.63 radians while traveling 1 mile.
Solution
To determine the angle \(\theta\) (in radians) that a tire turns while traveling 1 mile, follow these steps:
1. **Calculate the Circumference of the Tire:**
- **Diameter (\(D\))**: 37.64 inches
- **Circumference (\(C\))**: \( C = \pi \times D = \pi \times 37.64 \approx 118.12 \) inches
2. **Convert Travel Distance to Inches:**
- **1 mile** = 5,280 feet = 63,360 inches
3. **Determine the Number of Rotations:**
- **Number of rotations** = \(\frac{\text{Distance}}{\text{Circumference}} = \frac{63,360}{118.12} \approx 536.63\) rotations
4. **Calculate the Total Angle in Radians:**
- **Each rotation** corresponds to \(2\pi\) radians.
- **Total angle** (\(\theta\)) = Number of rotations \(\times 2\pi = 536.63 \times 2\pi \approx 3366.63\) radians
Thus, the tire turns approximately **3366.63 radians** while traveling 1 mile.
\[
\boxed{3366.63}
\]
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Simplify this solution Mind Expander
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.
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