Question

A target has a bull's-eye with a diameter of 2 inches. The outer ring is 1 inch wide. What is the area of the outer ring?

Answer 1

To find the area of the outer ring of the target, we first need to determine the area of the bull's-eye and the area of the entire target.

1. **Calculate the radius of the bull's-eye:**
The diameter of the bull's-eye is 2 inches, so the radius \( r_1 \) is:
\[
r_1 = \frac{2 \text{ inches}}{2} = 1 \text{ inch}
\]

2. **Calculate the radius of the entire target:**
The outer ring is 1 inch wide, which means we need to add that width to the radius of the bull's-eye. Thus, the radius \( r_2 \) of the entire target is:
\[
r_2 = r_1 + 1 \text{ inch} = 1 \text{ inch} + 1 \text{ inch} = 2 \text{ inches}
\]

3. **Calculate the area of the bull's-eye:**
The area \( A_1 \) of the bull's-eye is given by the formula for the area of a circle, \( A = \pi r^2 \):
\[
A_1 = \pi (r_1)^2 = \pi (1 \text{ inch})^2 = \pi \text{ square inches}
\]

4. **Calculate the area of the entire target:**
The area \( A_2 \) of the entire target is:
\[
A_2 = \pi (r_2)^2 = \pi (2 \text{ inches})^2 = \pi (4 \text{ square inches}) = 4\pi \text{ square inches}
\]

5. **Calculate the area of the outer ring:**
The area of the outer ring \( A_{\text{outer}} \) is the area of the entire target minus the area of the bull's-eye:
\[
A_{\text{outer}} = A_2 - A_1 = 4\pi - \pi = 3\pi \text{ square inches}
\]

Thus, the area of the outer ring is:
\[
\boxed{3\pi} \text{ square inches}
\]