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

Answer

To solve the problem step by step, we need to find the radii of two circles that touch each other externally, given that the sum of their areas is \(58\pi \, \text{cm}^2\) and the distance between their centers is \(10 \, \text{cm}\).

### Step 1: Define Variables
Let the radius of the first circle be \(r_1\) and the radius of the second circle be \(r_2\).

### Step 2: Set Up Equations
Since the circles touch externally, the distance between their centers is equal to the sum of their radii:
\[ r_1 + r_2 = 10 \quad \text{(1)} \]
The area of a circle is given by the formula \(A = \pi r^2\). Therefore, the sum of the areas of the two circles is:
\[ \pi r_1^2 + \pi r_2^2 = 58\pi \]
Dividing through by \(\pi\):
\[ r_1^2 + r_2^2 = 58 \quad \text{(2)} \]

### Step 3: Substitute Equation (1) into Equation (2)
From equation (1), we can express \(r_1\) in terms of \(r_2\):
\[ r_1 = 10 - r_2 \]
Now substitute this into equation (2):
\[ (10 - r_2)^2 + r_2^2 = 58 \]

### Step 4: Expand and Simplify
Expanding \((10 - r_2)^2\):
\[ 100 - 20r_2 + r_2^2 + r_2^2 = 58 \]
Combine like terms:
\[ 100 - 20r_2 + 2r_2^2 = 58 \]
Rearranging gives:
\[ 2r_2^2 - 20r_2 + 100 - 58 = 0 \]
\[ 2r_2^2 - 20r_2 + 42 = 0 \]
Dividing the entire equation by 2:
\[ r_2^2 - 10r_2 + 21 = 0 \quad \text{(3)} \]

### Step 5: Solve the Quadratic Equation
Now we will solve the quadratic equation (3) using the quadratic formula:
\[ r_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -10\), and \(c = 21\):
\[ r_2 = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 21}}{2 \cdot 1} \]
\[ r_2 = \frac{10 \pm \sqrt{100 - 84}}{2} \]
\[ r_2 = \frac{10 \pm \sqrt{16}}{2} \]
\[ r_2 = \frac{10 \pm 4}{2} \]
Calculating the two possible values:
1. \(r_2 = \frac{14}{2} = 7\)
2. \(r_2 = \frac{6}{2} = 3\)

### Step 6: Find \(r_1\) for Each \(r_2\)
Using equation (1):
1. If \(r_2 = 7\), then \(r_1 = 10 - 7 = 3\).
2. If \(r_2 = 3\), then \(r_1 = 10 - 3 = 7\).

### Conclusion
Thus, the radii of the two circles are:
\[ r_1 = 3 \, \text{cm}, \quad r_2 = 7 \, \text{cm} \quad \text{or} \quad r_1 = 7 \, \text{cm}, \quad r_2 = 3 \, \text{cm}. \]