A wheel has a radius of 18 inches. The distance, in inches, the wheel travels when it rotates through an angle of `(2)/(3)pi` radians is closest to which value?

Options

A. `45`
B. `37`
C. `13`
D. `11`

Answer

To find the distance that a wheel travels when it rotates through an angle of \( \frac{2}{3} \pi \) radians, we can use the formula for the length of an arc. The formula is given by:

\[ \text{Arc Length} = \theta \times r \]

where:
- \( \theta \) is the angle in radians,
- \( r \) is the radius of the circle.

### Step-by-step Solution:

1. **Identify the given values**:
- Radius \( r = 18 \) inches
- Angle \( \theta = \frac{2}{3} \pi \) radians

2. **Substitute the values into the arc length formula**:
\[ \text{Arc Length} = \left(\frac{2}{3} \pi\right) \times 18 \]

3. **Calculate the arc length**:
- First, multiply \( 18 \) by \( \frac{2}{3} \):
\[ 18 \times \frac{2}{3} = 12 \]
- Now, multiply by \( \pi \):
\[ \text{Arc Length} = 12 \pi \]

4. **Approximate the value of \( 12 \pi \)**:
- Using \( \pi \approx 3.14 \):
\[ 12 \pi \approx 12 \times 3.14 = 37.68 \]

5. **Determine the closest value**:
- The distance the wheel travels when it rotates through an angle of \( \frac{2}{3} \pi \) radians is approximately \( 37.68 \) inches.

### Final Answer:
The closest value to the distance the wheel travels is approximately **38 inches**.