Question
Circle O has radius of 24 units. Arc XY located on the circle has a central angle of 75°. What is the area of the associated sector, in square units? A. 10π B. 5π C. 51π D. 120π
Ask by Medina Higgins.
Mar 21,2025 21:00
UpStudy AI Solution
Tutor-Verified Answer
Answer
The area of the sector is 120π square units.
Solution
To find the area of the sector associated with arc XY, we can use the formula for the area of a sector:
\[
A = \frac{\theta}{360} \times \pi r^2
\]
where:
- \( A \) is the area of the sector,
- \( \theta \) is the central angle in degrees,
- \( r \) is the radius of the circle.
Given:
- \( r = 24 \) units,
- \( \theta = 75^{\circ} \).
Now, let's substitute the values into the formula step by step:
1. Calculate \( r^2 \):
\[
r^2 = 24^2
\]
2. Substitute \( \theta \) and \( r^2 \) into the area formula:
\[
A = \frac{75}{360} \times \pi \times (24^2)
\]
Now, let's perform the calculations. First, we will calculate \( 24^2 \) and then the area \( A \).
Calculate the value by following steps:
- step0: Calculate:
\(24^{2}\)
- step1: Evaluate the power:
\(576\)
Calculate or simplify the expression \( (75/360) * \pi * (24^2) \).
Calculate the value by following steps:
- step0: Calculate:
\(\left(\frac{75}{360}\right)\pi \times 24^{2}\)
- step1: Reduce the fraction:
\(\frac{5}{24}\pi \times 24^{2}\)
- step2: Multiply:
\(\frac{5\pi }{24}\times 24^{2}\)
- step3: Reduce the numbers:
\(5\pi \times 24\)
- step4: Multiply the terms:
\(120\pi \)
1. The calculation of \( 24^2 \) gives us \( 576 \).
2. Substituting this into the area formula, we find that the area \( A \) of the sector is \( 120\pi \) square units.
Thus, the area of the associated sector is \( 120\pi \) square units, which corresponds to option D.
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Simplify this solution The Deep Dive
To find the area of the sector associated with arc XY, we can use the formula for the area of a sector, which is given by:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.
In this case, \(\theta = 75^\circ\) and \(r = 24\) units. Plugging in these values:
\[
\text{Area} = \frac{75}{360} \times \pi \times 24^2
\]
Calculating \(24^2\):
\[
24^2 = 576
\]
Now substituting this back into the formula:
\[
\text{Area} = \frac{75}{360} \times \pi \times 576
\]
This simplifies to:
\[
\text{Area} = \frac{75 \times 576 \pi}{360}
\]
Now, simplifying \(\frac{75}{360}\):
\[
\frac{75}{360} = \frac{5}{24}
\]
So we have:
\[
\text{Area} = \frac{5}{24} \times 576 \pi
\]
Calculating \(\frac{5 \times 576}{24}\):
\[
576 \div 24 = 24 \quad \text{and thus} \quad 5 \times 24 = 120
\]
Finally, we have:
\[
\text{Area} = 120\pi
\]
Therefore, the area of the associated sector is \( \boxed{120\pi} \).
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