To determine the sector area created by the hands of a clock with a radius of 9 inches when the time is 4:00, we need to follow these steps:
1. Calculate the angle between the hands of the clock at 4:00:
Each hour on the clock represents 30 degrees. The time 4:00 means the minute hand is at the 12, and the hour hand is at the 4.
[tex]\[
\text{Angle} = 4 \times 30 = 120 \text{ degrees}
\][/tex]
2. Convert the angle from degrees to radians:
Degrees can be converted to radians using the conversion factor [tex]\(\pi \text{ radians} = 180 \text{ degrees}\)[/tex].
[tex]\[
\text{Angle in radians} = 120 \text{ degrees} \times \frac{\pi \text{ radians}}{180 \text{ degrees}} = \frac{120\pi}{180} = \frac{2\pi}{3} \text{ radians} \approx 2.0944 \text{ radians}
\][/tex]
3. Calculate the sector area using the formula:
The formula for the area of a sector of a circle is:
[tex]\[
A = \frac{1}{2} \times r^2 \times \theta
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( \theta \)[/tex] is the angle in radians.
[tex]\[
A = \frac{1}{2} \times 9^2 \times \frac{2\pi}{3}
\][/tex]
Breaking it down:
[tex]\[
r^2 = 9^2 = 81
\][/tex]
Therefore:
[tex]\[
A = \frac{1}{2} \times 81 \times \frac{2\pi}{3}
\][/tex]
Simplifying further:
[tex]\[
A = \frac{81}{2} \times \frac{2\pi}{3} = \frac{81 \times 2\pi}{2 \times 3} = \frac{81 \times 2\pi}{6} = \frac{81\pi}{3} = 27\pi \text{ square inches}
\][/tex]
So, the sector area created by the hands of the clock at 4:00 with a radius of 9 inches is:
[tex]\[
\boxed{27 \pi \text{ in}^{2}}
\][/tex]
None of the other choices [tex]\((6.75 \pi \text{ in}^2, 20.25 \pi \text{ in}^2, 81 \pi \text{ in}^2)\)[/tex] match this calculation. Thus, the correct choice is [tex]\(27 \pi \text{ in}^2\)[/tex].
