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What is the sector area created by the hands of a clock with a radius of 9 inches when the time is [tex]4:00[/tex]?

A. [tex]6.75 \pi \, \text{in}^2[/tex]
B. [tex]20.25 \pi \, \text{in}^2[/tex]
C. [tex]27 \pi \, \text{in}^2[/tex]
D. [tex]81 \pi \, \text{in}^2[/tex]


Sagot :

To determine the sector area created by the hands of a clock with a radius of 9 inches when the time is 4:00, we need to follow these steps:

1. Calculate the angle between the hands of the clock at 4:00:

Each hour on the clock represents 30 degrees. The time 4:00 means the minute hand is at the 12, and the hour hand is at the 4.

[tex]\[ \text{Angle} = 4 \times 30 = 120 \text{ degrees} \][/tex]

2. Convert the angle from degrees to radians:

Degrees can be converted to radians using the conversion factor [tex]\(\pi \text{ radians} = 180 \text{ degrees}\)[/tex].

[tex]\[ \text{Angle in radians} = 120 \text{ degrees} \times \frac{\pi \text{ radians}}{180 \text{ degrees}} = \frac{120\pi}{180} = \frac{2\pi}{3} \text{ radians} \approx 2.0944 \text{ radians} \][/tex]

3. Calculate the sector area using the formula:

The formula for the area of a sector of a circle is:

[tex]\[ A = \frac{1}{2} \times r^2 \times \theta \][/tex]

where [tex]\( r \)[/tex] is the radius and [tex]\( \theta \)[/tex] is the angle in radians.

[tex]\[ A = \frac{1}{2} \times 9^2 \times \frac{2\pi}{3} \][/tex]

Breaking it down:

[tex]\[ r^2 = 9^2 = 81 \][/tex]

Therefore:

[tex]\[ A = \frac{1}{2} \times 81 \times \frac{2\pi}{3} \][/tex]

Simplifying further:

[tex]\[ A = \frac{81}{2} \times \frac{2\pi}{3} = \frac{81 \times 2\pi}{2 \times 3} = \frac{81 \times 2\pi}{6} = \frac{81\pi}{3} = 27\pi \text{ square inches} \][/tex]

So, the sector area created by the hands of the clock at 4:00 with a radius of 9 inches is:

[tex]\[ \boxed{27 \pi \text{ in}^{2}} \][/tex]

None of the other choices [tex]\((6.75 \pi \text{ in}^2, 20.25 \pi \text{ in}^2, 81 \pi \text{ in}^2)\)[/tex] match this calculation. Thus, the correct choice is [tex]\(27 \pi \text{ in}^2\)[/tex].
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