To find the area of the sector created by the hands of a clock at 4:00, follow these steps:
1. Identify the angle formed by the hands at 4:00:
- The clock is divided into 12 hours.
- Each hour creates an angle of [tex]\( \frac{360^\circ}{12} = 30^\circ \)[/tex].
- At 4:00, the hour hand is on 4 and the minute hand is on 12.
- The angle between 4 and 12 is [tex]\( 4 \times 30^\circ = 120^\circ \)[/tex].
2. Convert the angle from degrees to radians:
- There are [tex]\(2\pi\)[/tex] radians in a full circle (360 degrees).
- Therefore, [tex]\(120^\circ\)[/tex] in radians is [tex]\( \frac{120^\circ}{360^\circ} \times 2\pi = \frac{1}{3}\times 2\pi = \frac{2\pi}{3}\)[/tex] radians.
3. Calculate the area of the sector:
- The formula for the area of a sector is [tex]\(\frac{1}{2} \times r^2 \times \theta \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( \theta \)[/tex] is the angle in radians.
- Given the radius [tex]\( r = 9 \)[/tex] inches and [tex]\( \theta = \frac{2\pi}{3} \)[/tex] radians:
[tex]\[
\text{Sector area} = \frac{1}{2} \times 9^2 \times \frac{2\pi}{3}
\][/tex]
[tex]\[
= \frac{1}{2} \times 81 \times \frac{2\pi}{3}
\][/tex]
[tex]\[
= \frac{1}{2} \times 81 \times \frac{2\pi}{3}
\][/tex]
[tex]\[
= \frac{81 \times 2\pi}{6}
\][/tex]
[tex]\[
= \frac{162\pi}{6}
\][/tex]
[tex]\[
= 27\pi
\][/tex]
So, the area of the sector is [tex]\( 27\pi \)[/tex] square inches.
Given options were:
- [tex]\( 6.75\pi \)[/tex] in. [tex]\( ^2 \)[/tex]
- [tex]\( 20.25\pi \)[/tex] in. [tex]\( ^2 \)[/tex]
- [tex]\( 27\pi \)[/tex] in. [tex]\( ^2 \)[/tex]
- [tex]\( 81\pi \)[/tex] in. [tex]\( ^2 \)[/tex]
The correct answer is:
- [tex]\( 27\pi \)[/tex] in. [tex]\( ^2 \)[/tex]
