The diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 24 cm and 7 cm is

Options

A. 31 cm
B. 25 cm
C. 62 cm
D. 50 cm

Explanation

To find the diameter of a circle whose area is equal to the sum of the areas of two circles with radii 24 cm and 7 cm, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Given Circles:**
- Let the radius of the first circle (r1) be 24 cm.
- Let the radius of the second circle (r2) be 7 cm.

2. **Calculate the Areas of the Two Circles:**
- The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
- For the first circle:
\[ \text{Area}_1 = \pi (24)^2 = \pi \times 576 \]
- For the second circle:
\[ \text{Area}_2 = \pi (7)^2 = \pi \times 49 \]

3. **Sum the Areas of the Two Circles:**
- The total area (A) of the two circles is:
\[ A = \text{Area}_1 + \text{Area}_2 = \pi \times 576 + \pi \times 49 = \pi (576 + 49) = \pi \times 625 \]

4. **Set Up the Equation for the New Circle:**
- Let the radius of the new circle be r. The area of this circle can also be expressed as:
\[ \text{Area} = \pi r^2 \]
- According to the problem, this area is equal to the sum of the areas of the two circles:
\[ \pi r^2 = \pi \times 625 \]

5. **Cancel π from Both Sides:**
- Dividing both sides by π, we get:
\[ r^2 = 625 \]

6. **Solve for r:**
- Taking the square root of both sides:
\[ r = \sqrt{625} = 25 \text{ cm} \]

7. **Calculate the Diameter:**
- The diameter (D) of a circle is given by:
\[ D = 2r \]
- Therefore:
\[ D = 2 \times 25 = 50 \text{ cm} \]

### Final Answer:
The diameter of the circle is **50 cm**.

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