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 can follow these steps:

### Step 1: Calculate the area of the first circle (radius = 24 cm)
The formula for the area of a circle is given by:
\[ \text{Area} = \pi r^2 \]
For the first circle:
\[ \text{Area}_1 = \pi (24)^2 = \pi \times 576 \text{ cm}^2 \]

### Step 2: Calculate the area of the second circle (radius = 7 cm)
Using the same formula for the second circle:
\[ \text{Area}_2 = \pi (7)^2 = \pi \times 49 \text{ cm}^2 \]

### Step 3: Find the sum of the areas of the two circles
Now, we add the areas of both circles:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = \pi \times 576 + \pi \times 49 = \pi (576 + 49) = \pi \times 625 \text{ cm}^2 \]

### Step 4: Set the total area equal to the area of the new circle
Let the radius of the new circle be \( r \). The area of this circle can also be expressed as:
\[ \pi r^2 \]
Setting the areas equal gives us:
\[ \pi r^2 = \pi \times 625 \]

### Step 5: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ r^2 = 625 \]

### Step 6: Solve for \( r \)
Taking the square root of both sides:
\[ r = \sqrt{625} = 25 \text{ cm} \]

### Step 7: Calculate the diameter of the new circle
The diameter \( d \) is twice the radius:
\[ d = 2r = 2 \times 25 = 50 \text{ cm} \]

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

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