https://file.aiquickdraw.com/mathbot/file/1729587830315i6krgmtg.jpg

Let's work through the two problems step by step.

### Problem 1:
"The area of a sector of a circle with a radius measuring \( 15 \, \text{cm} \) is \( 75\pi \, \text{cm}^2 \). What is the measure of the central angle that forms the sector?"

#### Formula:
The area \( A \) of a sector is given by:
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
where:
- \( A \) is the area of the sector,
- \( \theta \) is the central angle (in degrees),
- \( r \) is the radius of the circle.

Substitute the known values:
\[
75\pi = \frac{\theta}{360^\circ} \times \pi (15)^2
\]
Simplifying:
\[
75\pi = \frac{\theta}{360^\circ} \times \pi \times 225
\]
Cancel out \( \pi \) from both sides:
\[
75 = \frac{\theta}{360^\circ} \times 225
\]
Solving for \( \theta \):
\[
75 \times 360 = \theta \times 225
\]
\[
27000 = 225 \theta
\]
\[
\theta = \frac{27000}{225} = 120^\circ
\]

So, the central angle is \( 120^\circ \).

### Problem 2:
"In a circle with a diameter of 32, the area of a sector is \( \frac{512\pi}{3} \). The measure of the angle of the sector, in radians, is?"

#### Step-by-step solution:

1. The radius \( r \) is half the diameter:
   \[
   r = \frac{32}{2} = 16
   \]

2. The area of the entire circle is:
   \[
   A_{\text{circle}} = \pi r^2 = \pi (16)^2 = 256\pi
   \]

3. The area of a sector is given by:
   \[
   A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2
   \]
   Simplifying:
   \[
   A_{\text{sector}} = \frac{\theta}{2\pi} \times 256\pi
   \]
   which becomes:
   \[
   A_{\text{sector}} = \frac{\theta}{2} \times 256
   \]

4. Set the area of the sector equal to the given value \( \frac{512\pi}{3} \):
   \[
   \frac{\theta}{2} \times 256 = \frac{512\pi}{3}
   \]

5. Solve for \( \theta \):
   \[
   \frac{\theta \times 256}{2} = \frac{512\pi}{3}
   \]
   Multiply both sides by 2:
   \[
   \theta \times 256 = \frac{1024\pi}{3}
   \]

6. Divide by 256:
   \[
   \theta = \frac{1024\pi}{3 \times 256}
   \]
   Simplifying:
   \[
   \theta = \frac{4\pi}{3}
   \]

Thus, the central angle is \( \frac{4\pi}{3} \) radians.

---

Would you like further details or explanations on these? Here are some related questions to extend your understanding:

1. How do we derive the formula for the area of a sector?
2. Can you find the central angle in radians instead of degrees for the first problem?
3. How is the formula for the area of a sector different when using radians instead of degrees?
4. If the radius was doubled, how would that affect the area of the sector?
5. How would you convert between radians and degrees for central angles?

**Tip:** In problems involving sectors, pay attention to the units of the angle (degrees vs radians) since it affects the formula used.

The area of a sector of a circle with a radius measuring 15 cm is 75π cm². What is the measure of the central angle that forms the sector? And in another circle with a diameter of 32, the area of a sector is 512π/3. What is the measure of the angle of the sector, in radians?

Learn how to calculate the central angle of a sector of a circle with a radius of 15 cm and an area of 75π cm². Also, explore how to determine the angle in radians for a sector in a circle with a diameter of 32 and area 512π/3. This lesson covers key geometric concepts of sectors and angles, suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation