To determine the sector area created by the hands of a clock when the time is 4:00, let's go through the steps step-by-step:
1. Determine the Clock Angle:
The clock is divided into 12 hours, and each hour represents an angle of [tex]\(\frac{360}{12} = 30\)[/tex] degrees.
At 4:00, the minute hand points at the 12, and the hour hand points at the 4. This means the angle between the hour hand and the vertical 12 o'clock position is [tex]\(4 \times 30 = 120\)[/tex] degrees.
2. Convert the Angle to Radians:
To use the formula for the area of a sector, we need to convert degrees to radians. The conversion factor is [tex]\(\frac{\pi}{180}\)[/tex].
[tex]\[ 120 \text{ degrees} = 120 \times \frac{\pi}{180} = \frac{2\pi}{3} \text{ radians} \][/tex]
3. Apply the Sector Area Formula:
The formula for the area of a sector is [tex]\(\frac{1}{2} \theta r^2\)[/tex], where [tex]\(\theta\)[/tex] is the angle in radians, and [tex]\(r\)[/tex] is the radius.
Given:
- Radius [tex]\(r = 9\)[/tex] inches
- Angle [tex]\(\theta = \frac{2\pi}{3}\)[/tex] radians
Now,[tex]\[
\text{Area} = \frac{1}{2} \times \frac{2\pi}{3} \times (9^2)
\][/tex]
[tex]\[
= \frac{1}{2} \times \frac{2\pi}{3} \times 81
\][/tex]
[tex]\[
= \frac{\pi}{3} \times 81
\][/tex]
[tex]\[
= 27\pi \text{ square inches}
\][/tex]
Therefore, the sector area created by the hands of a clock with a radius of 9 inches when the time is 4:00 is
[tex]\[ 27\pi \text{ square inches} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{27\pi \text{ in}^2} \][/tex]
