Find the perimeter of a cirlce whose area is equal to sum of areas of the circles with diameters 10 cm and 24 cm. Give your answer correct to two decimal places.

Answer

To find the perimeter of a circle whose area is equal to the sum of the areas of two circles with diameters of 10 cm and 24 cm, we can follow these steps:

### Step 1: Calculate the radii of the two circles.
- For the first circle (diameter = 10 cm):
\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{10}{2} = 5 \text{ cm} \]

- For the second circle (diameter = 24 cm):
\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 2: Calculate the areas of the two circles.
- Area of the first circle (C1):
\[ \text{Area} = \pi r^2 = \pi (5^2) = \pi (25) = 25\pi \text{ cm}^2 \]

- Area of the second circle (C2):
\[ \text{Area} = \pi r^2 = \pi (12^2) = \pi (144) = 144\pi \text{ cm}^2 \]

### Step 3: Find the total area of the two circles.
\[ \text{Total Area} = \text{Area of C1} + \text{Area of C2} = 25\pi + 144\pi = 169\pi \text{ cm}^2 \]

### Step 4: Set the area of the new circle equal to the total area.
Let the radius of the new circle be \( r \). The area of the new circle is:
\[ \pi r^2 = 169\pi \]

### Step 5: Solve for \( r^2 \).
Dividing both sides by \( \pi \):
\[ r^2 = 169 \]

### Step 6: Find \( r \).
Taking the square root of both sides:
\[ r = \sqrt{169} = 13 \text{ cm} \]

### Step 7: Calculate the perimeter of the new circle.
The perimeter (circumference) of a circle is given by:
\[ \text{Perimeter} = 2\pi r = 2\pi(13) = 26\pi \text{ cm} \]

### Step 8: Calculate \( 26\pi \) to two decimal places.
Using \( \pi \approx 3.14 \):
\[ \text{Perimeter} \approx 26 \times 3.14 = 81.64 \text{ cm} \]

### Final Answer:
The perimeter of the circle is approximately **81.64 cm**.

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