The area (in `m^2`) of a circular path of uniform width x metres surrounding a circular region of diameter d metres is .....

Options

A. `pi x(x + 2d)`
B. `pix(x+d)`
C. `pix(2x+d)`
D. `pix(x+(d)/(2))`

Answer

To find the area of a circular path of uniform width \( x \) metres surrounding a circular region of diameter \( d \) metres, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Inner Circle's Radius:**
The diameter of the inner circle is given as \( d \) metres. Therefore, the radius \( r \) of the inner circle is:
\[ r = \frac{d}{2} \]

2. **Identify the Outer Circle's Radius:**
The outer circle surrounds the inner circle and has a uniform width \( x \). Thus, the radius \( R \) of the outer circle is:
\[ R = r + x = \frac{d}{2} + x \]

3. **Calculate the Area of the Outer Circle:**
The area \( A_{outer} \) of the outer circle can be calculated using the formula for the area of a circle, \( \pi R^2 \):
\[ A_{outer} = \pi R^2 = \pi \left(\frac{d}{2} + x\right)^2 \]

4. **Calculate the Area of the Inner Circle:**
The area \( A_{inner} \) of the inner circle is:
\[ A_{inner} = \pi r^2 = \pi \left(\frac{d}{2}\right)^2 \]

5. **Calculate the Area of the Circular Path:**
The area of the circular path, which is the area between the outer circle and the inner circle, is given by:
\[ A_{path} = A_{outer} - A_{inner} \]
Substituting the areas we calculated:
\[ A_{path} = \pi \left(\frac{d}{2} + x\right)^2 - \pi \left(\frac{d}{2}\right)^2 \]

6. **Simplify the Expression:**
Factor out \( \pi \):
\[ A_{path} = \pi \left[\left(\frac{d}{2} + x\right)^2 - \left(\frac{d}{2}\right)^2\right] \]
Now, expand the first term:
\[ \left(\frac{d}{2} + x\right)^2 = \left(\frac{d}{2}\right)^2 + 2\left(\frac{d}{2}\right)x + x^2 \]
Thus, we have:
\[ A_{path} = \pi \left[\left(\frac{d}{2}\right)^2 + 2\left(\frac{d}{2}\right)x + x^2 - \left(\frac{d}{2}\right)^2\right] \]
The \( \left(\frac{d}{2}\right)^2 \) terms cancel out:
\[ A_{path} = \pi \left[2\left(\frac{d}{2}\right)x + x^2\right] \]
Simplifying further:
\[ A_{path} = \pi \left[dx + x^2\right] \]

7. **Final Result:**
Therefore, the area of the circular path is:
\[ A_{path} = \pi x (x + d) \]