Find the diameter of the circle whose area is equal to the sum of the areas of two circles having radii 4 cm and 3 cm.

Explanation

To find the diameter of the circle whose area is equal to the sum of the areas of two circles with radii 4 cm and 3 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the area of the first circle (radius = 4 cm)**:
\[ \text{Area}_1 = \pi r_1^2 = \pi (4)^2 = 16\pi \text{ cm}^2 \]

2. **Calculate the area of the second circle (radius = 3 cm)**:
\[ \text{Area}_2 = \pi r_2^2 = \pi (3)^2 = 9\pi \text{ cm}^2 \]

3. **Find the sum of the areas of the two circles**:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = 16\pi + 9\pi = 25\pi \text{ cm}^2 \]

4. **Set the area of the new circle equal to the total area**:
\[ \text{Area of new circle} = \pi R^2 = 25\pi \]
Here, \( R \) is the radius of the new circle.

5. **Divide both sides by \(\pi\)** to simplify:
\[ R^2 = 25 \]

6. **Take the square root of both sides to find the radius \( R \)**:
\[ R = \sqrt{25} = 5 \text{ cm} \]

7. **Calculate the diameter \( D \) of the new circle**:
\[ D = 2R = 2 \times 5 = 10 \text{ cm} \]

Thus, the diameter of the circle is **10 cm**.