The radii of two circles are 15 cm and 8 cm.If the area of a third circle is equal to the sum of the areas of the two circles, then what will be the radius of the third circle?

Options

A. 17 cm
B. 23 cm
C. 21 cm
D. 19 cm

Answer

To find the radius of the third circle whose area is equal to the sum of the areas of two given circles, we can follow these steps:

### Step 1: Identify the radii of the two circles
Let the radius of the first circle (R1) be 15 cm and the radius of the second circle (R2) be 8 cm.

### Step 2: Write the formula for the area of a circle
The area (A) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 3: Calculate the areas of the first and second circles
Using the formula for the area:
- Area of the first circle (A1):
\[ A1 = \pi R1^2 = \pi (15^2) = \pi (225) \]
- Area of the second circle (A2):
\[ A2 = \pi R2^2 = \pi (8^2) = \pi (64) \]

### Step 4: Sum the areas of the two circles
Now, we can find the total area of the two circles:
\[ A1 + A2 = \pi (225) + \pi (64) = \pi (225 + 64) = \pi (289) \]

### Step 5: Set the area of the third circle equal to the sum of the areas
Let the radius of the third circle be R3. The area of the third circle (A3) is:
\[ A3 = \pi R3^2 \]
According to the problem, this area is equal to the sum of the areas of the first two circles:
\[ \pi R3^2 = \pi (289) \]

### Step 6: Simplify the equation
We can divide both sides by \( \pi \) (since \( \pi \) is a common factor):
\[ R3^2 = 289 \]

### Step 7: Solve for R3
To find R3, take the square root of both sides:
\[ R3 = \sqrt{289} = 17 \text{ cm} \]

### Conclusion
The radius of the third circle is 17 cm.

---