The bottom of a pendulum traces an arc 3 feet in length when the pendulum swings through an angle of `(1)/(2)` radians.What is the number of feet in the length of the pendulum?

Options

A. `1.5`
B. `6`
C. `(1.5)/(pi)`
D. `6pi`

Answer

To find the length of the pendulum, we can use the relationship between the angle in radians, the length of the arc, and the radius of the circle. The formula we will use is:

\[ \text{Angle} = \frac{\text{Arc Length}}{\text{Radius}} \]

### Step-by-step Solution:

1. **Identify the given values**:
- Arc Length (s) = 3 feet
- Angle (θ) = \(\frac{1}{2}\) radians

2. **Write the formula**:
\[ \theta = \frac{s}{r} \]
where:
- θ = angle in radians
- s = arc length
- r = radius (length of the pendulum)

3. **Rearrange the formula to solve for the radius (r)**:
\[ r = \frac{s}{\theta} \]

4. **Substitute the known values into the formula**:
\[ r = \frac{3 \text{ feet}}{\frac{1}{2} \text{ radians}} \]

5. **Calculate the radius**:
- Dividing by a fraction is the same as multiplying by its reciprocal:
\[ r = 3 \text{ feet} \times \frac{2}{1} = 6 \text{ feet} \]

6. **Conclusion**:
The length of the pendulum is 6 feet.

### Final Answer:
The length of the pendulum is **6 feet**.

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