Answer :
To determine the area of the sector created by the hands of a clock at 4:00, we can follow a detailed, step-by-step process:
### Step-by-Step Solution
1. Calculate the Angle Formed by the Clock Hands:
A clock is a circle divided into 12 hours. Each hour mark corresponds to an angle of [tex]\(30^\circ\)[/tex] (since [tex]\(360^\circ / 12 = 30^\circ\)[/tex]).
The angle for 4:00 is:
[tex]\[ 4 \text{ hours} \times 30^\circ \text{ per hour} = 120^\circ \][/tex]
2. Convert the Angle to Radians (if necessary):
To use it in area calculation, ensure the angle is in the right unit. In this case, the formula we're using can work with degrees directly, so no need for conversion.
3. Apply the Sector Area Formula:
The formula for the area of a sector of a circle is given by:
[tex]\[ A = \left( \frac{\theta}{360} \right) \times \pi \times r^2 \][/tex]
Where:
- [tex]\(\theta\)[/tex] is the angle of the sector in degrees.
- [tex]\(r\)[/tex] is the radius of the circle.
4. Substitute Known Values:
Here, [tex]\(\theta = 120^\circ\)[/tex] and [tex]\(r = 9 \text{ inches}\)[/tex].
So, the area [tex]\(A\)[/tex] is:
[tex]\[ A = \left( \frac{120}{360} \right) \times \pi \times 9^2 \][/tex]
5. Simplify the Expression:
Calculate the fraction for the angle:
[tex]\[ \frac{120}{360} = \frac{1}{3} \][/tex]
Then, substitute and simplify:
[tex]\[ A = \frac{1}{3} \times \pi \times 81 \][/tex]
[tex]\[ A = \frac{81}{3} \pi \][/tex]
[tex]\[ A = 27 \pi \text{ square inches} \][/tex]
### Final Answer
The area of the sector created by the hands of the clock at 4:00 with a radius of 9 inches is:
[tex]\[ 27 \pi \text{ square inches} \][/tex]
Hence, the correct answer of the given choices:
[tex]\[ \boxed{27 \pi \text{ in}^2} \][/tex]
### Step-by-Step Solution
1. Calculate the Angle Formed by the Clock Hands:
A clock is a circle divided into 12 hours. Each hour mark corresponds to an angle of [tex]\(30^\circ\)[/tex] (since [tex]\(360^\circ / 12 = 30^\circ\)[/tex]).
The angle for 4:00 is:
[tex]\[ 4 \text{ hours} \times 30^\circ \text{ per hour} = 120^\circ \][/tex]
2. Convert the Angle to Radians (if necessary):
To use it in area calculation, ensure the angle is in the right unit. In this case, the formula we're using can work with degrees directly, so no need for conversion.
3. Apply the Sector Area Formula:
The formula for the area of a sector of a circle is given by:
[tex]\[ A = \left( \frac{\theta}{360} \right) \times \pi \times r^2 \][/tex]
Where:
- [tex]\(\theta\)[/tex] is the angle of the sector in degrees.
- [tex]\(r\)[/tex] is the radius of the circle.
4. Substitute Known Values:
Here, [tex]\(\theta = 120^\circ\)[/tex] and [tex]\(r = 9 \text{ inches}\)[/tex].
So, the area [tex]\(A\)[/tex] is:
[tex]\[ A = \left( \frac{120}{360} \right) \times \pi \times 9^2 \][/tex]
5. Simplify the Expression:
Calculate the fraction for the angle:
[tex]\[ \frac{120}{360} = \frac{1}{3} \][/tex]
Then, substitute and simplify:
[tex]\[ A = \frac{1}{3} \times \pi \times 81 \][/tex]
[tex]\[ A = \frac{81}{3} \pi \][/tex]
[tex]\[ A = 27 \pi \text{ square inches} \][/tex]
### Final Answer
The area of the sector created by the hands of the clock at 4:00 with a radius of 9 inches is:
[tex]\[ 27 \pi \text{ square inches} \][/tex]
Hence, the correct answer of the given choices:
[tex]\[ \boxed{27 \pi \text{ in}^2} \][/tex]
