To determine the area of the sector created by the hands of a clock at 4:00 with a radius of 9 inches, we need to follow these steps:
1. Understand the Clock Geometry:
- A clock is a circle, and each hour represents a sector of the circle. The clock is divided into 12 hours, so each hour represents an angle of [tex]\( \frac{360^\circ}{12} = 30^\circ \)[/tex].
2. Calculate the Angle:
- At 4:00, the angle between the 12:00 mark and the 4:00 mark is [tex]\( 4 \times 30^\circ = 120^\circ \)[/tex].
3. Convert the Angle to Radians:
- To use the sector area formula, we need the angle in radians. The conversion factor from degrees to radians is [tex]\( \frac{\pi}{180^\circ} \)[/tex].
- Therefore, the angle in radians is [tex]\( 120^\circ \times \frac{\pi}{180^\circ} = \frac{120\pi}{180} = \frac{2\pi}{3} \)[/tex] radians.
4. Use the Sector Area Formula:
- The formula for the area of a sector is [tex]\( \frac{1}{2} r^2 \theta \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( \theta \)[/tex] is the angle in radians.
- Plugging in the values, [tex]\( r = 9 \)[/tex] inches and [tex]\( \theta = \frac{2\pi}{3} \)[/tex], we get:
[tex]\[
\text{Sector Area} = \frac{1}{2} \times 9^2 \times \frac{2\pi}{3}
\][/tex]
5. Simplify the Expression:
- First, calculate [tex]\( 9^2 \)[/tex]:
[tex]\[
9^2 = 81
\][/tex]
- Then, multiply by [tex]\( \frac{2\pi}{3} \)[/tex]:
[tex]\[
81 \times \frac{2\pi}{3} = 81 \times \frac{2}{3} \times \pi = 54 \pi
\][/tex]
- Finally, multiply by [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[
\frac{1}{2} \times 54 \pi = 27 \pi
\][/tex]
Therefore, the area of the sector created by the hands of a clock with a radius of 9 inches when the time is 4:00 is [tex]\( \boxed{27 \pi \, \text{in}^2} \)[/tex].
