https://file.aiquickdraw.com/mathbot/file/1730470862339ga7ynt1g.jpg

To solve the problem:

Given:
- Radius \( r = 23 \, \text{mm} \)
- Central angle \( \theta = \frac{3\pi}{16} \, \text{radians} \)

### Solution

#### Part (a): The Area of the Failed Sector

The formula for the area \( A \) of a sector is:
\[
A = \frac{1}{2} r^2 \theta
\]

Plugging in the values:
\[
A = \frac{1}{2} \times (23)^2 \times \frac{3\pi}{16}
\]

1. Calculate \( 23^2 = 529 \).
2. Substitute into the formula:
   \[
   A = \frac{1}{2} \times 529 \times \frac{3\pi}{16}
   \]
3. Simplify:
   \[
   A = \frac{529 \times 3\pi}{32} \approx 49.63 \, \text{mm}^2
   \]

#### Part (b): The Arc Length of the Failed Sector

The formula for the arc length \( L \) is:
\[
L = r \theta
\]

Plugging in the values:
\[
L = 23 \times \frac{3\pi}{16}
\]

1. Multiply:
   \[
   L = \frac{69\pi}{16} \approx 13.54 \, \text{mm}
   \]

### Answers:
- **(a)** The area of the failed sector is approximately \( 49.63 \, \text{mm}^2 \).
- **(b)** The arc length of the failed sector is approximately \( 13.54 \, \text{mm} \).

Would you like more details or have questions about these calculations?

#### Related Questions:
1. How do we calculate the area of a full circle?
2. What would the area be if the radius were doubled?
3. How does changing the angle affect the arc length?
4. How can we express these results in terms of pi?
5. Why do we use radians instead of degrees in these formulas?

#### Tip:
For sector problems, remember that radians simplify arc length and area formulas due to their direct relationship with the radius.

A hard disk drive has a failed sector with a radius of 23 mm and an angle of 3π/16 radians. Calculate the area and the arc length of the failed sector.

Learn how to calculate the area and arc length of a failed sector in a hard disk drive with a radius of 23 mm and an angle of 3π/16 radians. This problem covers key geometry concepts, including sector area and arc length formulas. Suitable for high school students in Grades 10-12.

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