To find the sector area created by the hands of a clock when the time is 4:00 and the radius of the clock is 9 inches, follow these steps:
1. Determine the angle (θ) in radians for the sector:
- The full circle represents 12 hours, and each hour marks an angle of [tex]\( \frac{2\pi}{12} = \frac{\pi}{6} \)[/tex] radians.
- At 4:00, we are 4 hours from the top (12:00) position, so the angle [tex]\( \theta = 4 \times \frac{\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \)[/tex] radians.
2. Calculate the area of the sector using the formula:
[tex]\[
\text{Sector Area} = \frac{1}{2} \times \text{radius}^2 \times \theta
\][/tex]
- Here, the radius [tex]\( r \)[/tex] is 9 inches and [tex]\( \theta \)[/tex] is [tex]\( \frac{2\pi}{3} \)[/tex] radians.
- Plugging these values into the formula:
[tex]\[
\text{Sector Area} = \frac{1}{2} \times 9^2 \times \frac{2\pi}{3}
\][/tex]
- Simplify the calculation:
[tex]\[
\text{Sector Area} = \frac{1}{2} \times 81 \times \frac{2\pi}{3} = \frac{81 \times 2\pi}{6} = \frac{162\pi}{6} = 27\pi \text{ square inches}
\][/tex]
Thus, the sector area created by the hands of a clock with a radius of 9 inches at 4:00 is:
[tex]\[ \boxed{27 \pi \text{ in}^2} \][/tex]
