At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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].
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].
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.