Answer :
To find the sector area created by the hands of a clock when the time is 4:00 and the radius of the clock is 9 inches, we need to follow these steps:
1. Understand the division of the clock face:
A clock is divided into 12 equal sectors (one for each hour). Each sector represents an angle of [tex]\( \frac{360^{\circ}}{12} = 30^{\circ} \)[/tex].
2. Calculate the angle at 4:00:
At 4:00, the angle between the hands is 4 sectors. Therefore:
[tex]\[ \text{Angle} = 4 \times 30^{\circ} = 120^{\circ} \][/tex]
3. Convert the angle from degrees to radians:
The formula for converting degrees to radians is:
[tex]\[ \text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right) \][/tex]
For [tex]\( 120^{\circ} \)[/tex]:
[tex]\[ 120^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{120 \pi}{180} = \frac{2\pi}{3} \text{ radians} \][/tex]
4. Use the sector area formula:
The formula for the area of a sector with radius [tex]\(r\)[/tex] and angle in radians [tex]\(\theta\)[/tex] is:
[tex]\[ \text{Sector Area} = \frac{1}{2} r^2 \theta \][/tex]
Plugging in the values [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]
Simplify the expression:
[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]\[ = 27\pi \text{ square inches} \][/tex]
Therefore, the sector area created by the hands of the clock at 4:00 with a radius of 9 inches is:
[tex]\[ 27 \pi \text{ square inches} \][/tex]
So the correct answer is:
[tex]\[ \boxed{27 \pi \text{ in}^2} \][/tex]
1. Understand the division of the clock face:
A clock is divided into 12 equal sectors (one for each hour). Each sector represents an angle of [tex]\( \frac{360^{\circ}}{12} = 30^{\circ} \)[/tex].
2. Calculate the angle at 4:00:
At 4:00, the angle between the hands is 4 sectors. Therefore:
[tex]\[ \text{Angle} = 4 \times 30^{\circ} = 120^{\circ} \][/tex]
3. Convert the angle from degrees to radians:
The formula for converting degrees to radians is:
[tex]\[ \text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right) \][/tex]
For [tex]\( 120^{\circ} \)[/tex]:
[tex]\[ 120^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{120 \pi}{180} = \frac{2\pi}{3} \text{ radians} \][/tex]
4. Use the sector area formula:
The formula for the area of a sector with radius [tex]\(r\)[/tex] and angle in radians [tex]\(\theta\)[/tex] is:
[tex]\[ \text{Sector Area} = \frac{1}{2} r^2 \theta \][/tex]
Plugging in the values [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]
Simplify the expression:
[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]\[ = 27\pi \text{ square inches} \][/tex]
Therefore, the sector area created by the hands of the clock at 4:00 with a radius of 9 inches is:
[tex]\[ 27 \pi \text{ square inches} \][/tex]
So the correct answer is:
[tex]\[ \boxed{27 \pi \text{ in}^2} \][/tex]
