Question
In a circle, a \( 180^{\circ} \) sector has area \( 98 \pi \mathrm{in}^{2} \). What is the radius of the circle? The radius of the circle is \( \square \mathrm{in} \). (Simplify your answer.)
Ask by Montgomery Bradley. in the United States
Mar 06,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The radius of the circle is 14 inches.
Solution
```markdown
**Step 1: Recognize the Relationship**
- A \(180^\circ\) sector represents half of a full circle.
- The area of the entire circle is given by \( \pi r^2 \).
- Therefore, the area of a \(180^\circ\) sector is:
\[
\frac{1}{2} \pi r^2
\]
**Step 2: Set Up the Equation**
- We are told that the area of the \(180^\circ\) sector is \( 98\pi \) in\(^2\).
- This gives us the equation:
\[
\frac{1}{2} \pi r^2 = 98\pi
\]
**Step 3: Solve for \( r \)**
- Multiply both sides of the equation by \(2\) to eliminate the fraction:
\[
\pi r^2 = 196\pi
\]
- Divide both sides by \(\pi\):
\[
r^2 = 196
\]
- Take the square root of both sides:
\[
r = \sqrt{196} = 14
\]
**Final Answer:**
The radius of the circle is \( \boxed{14 \text{ in}} \).
```
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Simplify this solution Beyond the Answer
The area \( A \) of a sector of a circle is given by the formula: \[ A = \frac{\theta}{360} \times \pi r^2 \] where \( \theta \) is the angle in degrees, and \( r \) is the radius. For a \( 180^{\circ} \) sector, this simplifies to: \[ A = \frac{180}{360} \times \pi r^2 = \frac{1}{2} \pi r^2 \] We know the area is \( 98 \pi \), so we set up the equation: \[ \frac{1}{2} \pi r^2 = 98 \pi \] Dividing both sides by \( \pi \): \[ \frac{1}{2} r^2 = 98 \] Multiplying both sides by 2: \[ r^2 = 196 \] Taking the square root: \[ r = \sqrt{196} = 14 \] Thus, the radius of the circle is \( 14 \, \mathrm{in} \).
