The area of the circle with radius y is w. The difference between the areas of the bigger circle (with radius y) and that of the smaller circle (with radius x) is w. So `(x)/(y)` is equal to

Options

A. `sqrt(1-(w')/(w))`
B. `sqrt(1+(w')/(w))`
C. `sqrt(1+(w)/(w'))`
D. `sqrt(1-(w)/(w'))`

Answer

To solve the problem, we need to find the ratio \(\frac{x}{y}\) given the areas of two circles.

### Step-by-Step Solution:

1. **Understand the Area of a Circle**:
The area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \(r\) is the radius of the circle.

2. **Calculate the Area of the Bigger Circle**:
For the bigger circle with radius \(y\), the area \(A_y\) is:
\[ A_y = \pi y^2 \]

3. **Calculate the Area of the Smaller Circle**:
For the smaller circle with radius \(x\), the area \(A_x\) is:
\[ A_x = \pi x^2 \]

4. **Set Up the Equation for the Difference in Areas**:
According to the problem, the difference between the areas of the bigger circle and the smaller circle is equal to \(w\):
\[ A_y - A_x = w \]
Substituting the areas we found:
\[ \pi y^2 - \pi x^2 = w \]

5. **Factor Out \(\pi\)**:
We can factor out \(\pi\) from the left side:
\[ \pi (y^2 - x^2) = w \]

6. **Rearranging the Equation**:
Dividing both sides by \(\pi\) gives:
\[ y^2 - x^2 = \frac{w}{\pi} \]

7. **Divide by \(y^2\)**:
Now, we divide the entire equation by \(y^2\):
\[ \frac{y^2 - x^2}{y^2} = \frac{w}{\pi y^2} \]
This simplifies to:
\[ 1 - \frac{x^2}{y^2} = \frac{w}{\pi y^2} \]

8. **Express \(\frac{x^2}{y^2}\)**:
Rearranging gives:
\[ \frac{x^2}{y^2} = 1 - \frac{w}{\pi y^2} \]

9. **Taking the Square Root**:
To find \(\frac{x}{y}\), we take the square root of both sides:
\[ \frac{x}{y} = \sqrt{1 - \frac{w}{\pi y^2}} \]

### Final Result:
Thus, the ratio \(\frac{x}{y}\) is:
\[ \frac{x}{y} = \sqrt{1 - \frac{w}{\pi y^2}} \]