Question

If the radii of two circles are 4 cm and 3 cm respectively, then the radius of the circle having area equal to the sum of the areas of these circles is:

• A. 5 cm
• B. 6 cm
• C. 7 cm
• D. 25 cm

Hint:

To find the radius of a circle whose area equals the sum of the areas of two circles, use the formula \( A = \pi r^2 \).

Answer 1

The Correct Option is A

The area of a circle is given by the formula: \[ A = \pi r^2 \] where \( r \) is the radius of the circle.
Step 1: Find the sum of the areas of the two circles.
For the first circle, with radius \( r_1 = 4 \) cm, the area is: \[ A_1 = \pi (4)^2 = 16\pi \, \text{sq. cm} \] For the second circle, with radius \( r_2 = 3 \) cm, the area is: \[ A_2 = \pi (3)^2 = 9\pi \, \text{sq. cm} \] Now, the sum of the areas of the two circles is: \[ A_{\text{total}} = A_1 + A_2 = 16\pi + 9\pi = 25\pi \, \text{sq. cm} \]
Step 2: Set the area of the new circle equal to the sum of the areas.
Let the radius of the new circle be \( r \). The area of the new circle is: \[ A_{\text{new}} = \pi r^2 \] Since the areas are equal: \[ \pi r^2 = 25\pi \]
Step 3: Solve for \( r \).
Dividing both sides by \( \pi \): \[ r^2 = 25 \] Taking the square root of both sides: \[ r = 5 \, \text{cm} \]
Step 4: Conclusion.
Therefore, the radius of the new circle is 5 cm.