To determine the sector area created by the hands of a clock with a radius of 9 inches when the time is 4:00, let's follow a systematic approach.
1. Determine the angle formed by the hands at 4:00:
- On a clock, each hour represents [tex]\(30\)[/tex] degrees since there are 12 hours on a clock and [tex]\(360\)[/tex] degrees in a full circle.
- At 4:00, the angle formed by the hour hand at the 4 o'clock mark is:
[tex]\[
\text{Angle} = 4 \times 30 = 120 \text{ degrees}
\][/tex]
2. Convert the angle from degrees to radians:
- The formula to convert degrees to radians is [tex]\(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)[/tex]:
[tex]\[
120^\circ \times \frac{\pi}{180} = \frac{2\pi}{3} \text{ radians}
\][/tex]
3. Calculate the sector area:
- The formula for the area of a sector is given by [tex]\(A = \frac{1}{2} r^2 \theta\)[/tex], where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is the central angle in radians.
- Substituting [tex]\(r = 9\)[/tex] inches and [tex]\(\theta = \frac{2\pi}{3}\)[/tex]:
[tex]\[
A = \frac{1}{2} \times 9^2 \times \frac{2\pi}{3}
\][/tex]
- Simplifying the expression step by step:
[tex]\[
9^2 = 81
\][/tex]
[tex]\[
A = \frac{1}{2} \times 81 \times \frac{2\pi}{3} = \frac{1}{2} \times 81 \times \frac{2 \pi}{3} = \frac{81 \times 2 \pi}{6} = \frac{162 \pi}{6} = 27\pi
\][/tex]
4. Conclusion:
- Therefore, the sector area formed by the clock hands at 4:00 is:
[tex]\[
27\pi \text{ in}^2
\][/tex]
So the correct option is:
[tex]\[
27 \pi \text{ in}^2
\][/tex]
