To find the area of a sector given the diameter and the angle in radians, you can use the following formula for the area of a sector:
\[ \text{Area} = \frac{1}{2} r^2 \theta \]
where \( r \) is the radius and \( \theta \) is the angle in radians.
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First, calculate the radius:
- The diameter is given as 16 ft, so the radius \( r \) is: \[ r = \frac{\text{diameter}}{2} = \frac{16 \text{ ft}}{2} = 8 \text{ ft} \]
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The angle \( \theta \) is given as \( \frac{\pi}{2} \) radians.
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Now, plug \( r \) and \( \theta \) into the area formula: \[ \text{Area} = \frac{1}{2} \times (8 \text{ ft})^2 \times \frac{\pi}{2} \] \[ = \frac{1}{2} \times 64 \text{ ft}^2 \times \frac{\pi}{2} \] \[ = \frac{64 \text{ ft}^2 \cdot \pi}{4} \] \[ = 16\pi \text{ ft}^2 \]
Thus, the area of the sector is \( 16\pi \text{ ft}^2 \).
If you need a numerical approximation, you can calculate \( 16\pi \approx 16 \times 3.14 = 50.24 \text{ ft}^2 \).
So, the area of the sector is approximately \( 50.24 \text{ ft}^2 \).