The area of a circular ring enclosed between two concentric circles is 286 cm? Find the radii of the two circles, given that their difference is 7 cm.

Answer

To solve the problem of finding the radii of two concentric circles given that the area of the circular ring between them is 286 cm² and the difference in their radii is 7 cm, we can follow these steps:

### Step-by-step Solution:

1. **Define Variables:**
Let \( R \) be the radius of the outer circle and \( r \) be the radius of the inner circle.

2. **Set Up the Equations:**
We know from the problem that:
\[ R - r = 7 \quad \text{(Equation 1)} \]
The area of the circular ring is given by the difference in the areas of the two circles:
\[ \pi R^2 - \pi r^2 = 286 \]
This can be simplified to:
\[ \pi (R^2 - r^2) = 286 \]

3. **Use the Difference of Squares:**
We can use the difference of squares formula \( a^2 - b^2 = (a + b)(a - b) \):
\[ R^2 - r^2 = (R + r)(R - r) \]
Substituting this into our area equation gives:
\[ \pi (R + r)(R - r) = 286 \]
Substituting \( R - r = 7 \) from Equation 1:
\[ \pi (R + r)(7) = 286 \]

4. **Solve for \( R + r \):**
Dividing both sides by \( 7 \):
\[ \pi (R + r) = \frac{286}{7} \]
Now, substituting \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} (R + r) = \frac{286}{7} \]
Multiplying both sides by \( 7 \):
\[ 22(R + r) = 286 \]
Dividing both sides by \( 22 \):
\[ R + r = \frac{286}{22} = 13 \quad \text{(Equation 2)} \]

5. **Solve the System of Equations:**
Now we have two equations:
\[ R - r = 7 \quad \text{(Equation 1)} \]
\[ R + r = 13 \quad \text{(Equation 2)} \]
Adding these two equations:
\[ (R - r) + (R + r) = 7 + 13 \]
This simplifies to:
\[ 2R = 20 \implies R = 10 \]

6. **Find \( r \):**
Substitute \( R = 10 \) back into Equation 1:
\[ 10 - r = 7 \implies r = 3 \]

7. **Final Answer:**
The radius of the outer circle \( R \) is 10 cm, and the radius of the inner circle \( r \) is 3 cm.

### Summary:
- Radius of the outer circle \( R = 10 \) cm
- Radius of the inner circle \( r = 3 \) cm