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
31 cm
25 cm
62 cm
50 cm
Step 1: Formula for the area of a circle is \(A = \pi r^2\), where \(r\) is the radius (in cm) and area is in square centimetres (cm²).
Step 2: First circle has radius \(24\,\text{cm}\). Its area is:
\(A_1 = \pi (24)^2 = \pi \times 576 = 576\pi\,\text{cm}^2\)
Step 3: Second circle has radius \(7\,\text{cm}\). Its area is:
\(A_2 = \pi (7)^2 = \pi \times 49 = 49\pi\,\text{cm}^2\)
Step 4: Add the two areas to get the total area:
\(A_{\text{total}} = A_1 + A_2 = 576\pi + 49\pi = 625\pi\,\text{cm}^2\)
Step 5: Let the radius of the required circle be \(R\,\text{cm}\). Its area should also be equal to \(625\pi\,\text{cm}^2\).
So, \(\pi R^2 = 625\pi\)
Step 6: Cancel \(\pi\) on both sides:
\(R^2 = 625\)
Step 7: Find \(R\) by taking square root:
\(R = \sqrt{625} = 25\,\text{cm}\)
Step 8: Diameter of a circle is twice the radius:
\(D = 2R = 2 \times 25 = 50\,\text{cm}\)
Final Answer: The diameter of the circle is 50 cm. So, the correct option is (D).