Answer :
To determine the sector area created by the hands of a clock at 1:00 when the clock has a radius of 9 inches, we need to consider the fraction of the circle that this time represents and then calculate the corresponding area.
1. Fraction of the Circle:
The hands of the clock create a sector that represents 1/12 of the entire clock face, since there are 12 hours on a clock.
2. Area of the Full Circle:
The area \(A\) of a full circle can be found using the formula:
[tex]\[ A = \pi r^2 \][/tex]
where \(r\) is the radius. Given that the radius \(r = 9\) inches:
[tex]\[ A = \pi \times 9^2 = \pi \times 81 = 81\pi \, \text{in}^2 \][/tex]
3. Sector Area:
Since the sector area is 1/12 of the entire circle's area, we multiply the total area by this fraction:
[tex]\[ \text{Sector Area} = \frac{1}{12} \times 81\pi = \frac{81\pi}{12} = 6.75\pi \, \text{in}^2 \][/tex]
Therefore, the sector area created by the hands of a clock with a radius of 9 inches at 1:00 is
[tex]\[ \boxed{6.75 \pi \, \text{in}^2} \][/tex]
1. Fraction of the Circle:
The hands of the clock create a sector that represents 1/12 of the entire clock face, since there are 12 hours on a clock.
2. Area of the Full Circle:
The area \(A\) of a full circle can be found using the formula:
[tex]\[ A = \pi r^2 \][/tex]
where \(r\) is the radius. Given that the radius \(r = 9\) inches:
[tex]\[ A = \pi \times 9^2 = \pi \times 81 = 81\pi \, \text{in}^2 \][/tex]
3. Sector Area:
Since the sector area is 1/12 of the entire circle's area, we multiply the total area by this fraction:
[tex]\[ \text{Sector Area} = \frac{1}{12} \times 81\pi = \frac{81\pi}{12} = 6.75\pi \, \text{in}^2 \][/tex]
Therefore, the sector area created by the hands of a clock with a radius of 9 inches at 1:00 is
[tex]\[ \boxed{6.75 \pi \, \text{in}^2} \][/tex]
