Two circles touch each other externally. The sum of their areas is `58pi cm^(2)` and the distance between their centres is 10 cm. Find the radii of the two circles.

Answer

To solve the problem, we need to find the radii of two circles that touch each other externally, given the sum of their areas and the distance between their centers.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The sum of the areas of the two circles is \( 58\pi \, \text{cm}^2 \).
- The distance between their centers is \( 10 \, \text{cm} \).

2. **Set Up the Equations:**
- Let the radius of the first circle be \( r_1 \) and the radius of the second circle be \( r_2 \).
- The area of the first circle is \( \pi r_1^2 \) and the area of the second circle is \( \pi r_2^2 \).
- Therefore, we can write the equation for the sum of the areas:
\[ \pi r_1^2 + \pi r_2^2 = 58\pi \]
- Dividing through by \( \pi \):
\[ r_1^2 + r_2^2 = 58 \quad \text{(Equation 1)} \]

3. **Use the Distance Between Centers:**
- Since the circles touch each other externally, the distance between their centers is equal to the sum of their radii:
\[ r_1 + r_2 = 10 \quad \text{(Equation 2)} \]

4. **Express \( r_1 \) in Terms of \( r_2 \):**
- From Equation 2, we can express \( r_1 \):
\[ r_1 = 10 - r_2 \]

5. **Substitute \( r_1 \) in Equation 1:**
- Substitute \( r_1 \) in Equation 1:
\[ (10 - r_2)^2 + r_2^2 = 58 \]
- Expanding \( (10 - r_2)^2 \):
\[ 100 - 20r_2 + r_2^2 + r_2^2 = 58 \]
- Combine like terms:
\[ 2r_2^2 - 20r_2 + 100 = 58 \]

6. **Simplify the Equation:**
- Rearranging gives:
\[ 2r_2^2 - 20r_2 + 42 = 0 \]
- Dividing the entire equation by 2:
\[ r_2^2 - 10r_2 + 21 = 0 \]

7. **Solve the Quadratic Equation:**
- Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1, b = -10, c = 21 \):
\[ r_2 = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 21}}{2 \cdot 1} \]
\[ r_2 = \frac{10 \pm \sqrt{100 - 84}}{2} \]
\[ r_2 = \frac{10 \pm \sqrt{16}}{2} \]
\[ r_2 = \frac{10 \pm 4}{2} \]
- This gives two possible values for \( r_2 \):
\[ r_2 = \frac{14}{2} = 7 \quad \text{or} \quad r_2 = \frac{6}{2} = 3 \]

8. **Determine the Values of \( r_1 \) and \( r_2 \):**
- If \( r_2 = 3 \), then:
\[ r_1 = 10 - r_2 = 10 - 3 = 7 \]
- If \( r_2 = 7 \), then:
\[ r_1 = 10 - r_2 = 10 - 7 = 3 \]
- Thus, the radii of the two circles are \( r_1 = 7 \, \text{cm} \) and \( r_2 = 3 \, \text{cm} \).

### Final Answer:
The radii of the two circles are \( 7 \, \text{cm} \) and \( 3 \, \text{cm} \).

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