Question

The area of circle is equal to the sum of the areas of two circles of radii 24 cm and 7 cm. The diameter of the new circle is
(a) 25 cm
(b) 31 cm
(c) 50 cm
(d) 62 cm

Answer 1

(c) 50 cm
Let r cm be the radius of the new circle.
Now,
Area of the new circle = Area of the circle with radius 24 cm + Area of the circle with radius 7 cm
Thus, we have:
$\pi r^2=\pi {r_1}^2+\pi {r_2}^2$
$$\begin{gathered} \Rightarrow \mathrm{\pi }{r}^{\mathit{2}}=\left[\mathrm{\pi }\times {\left(24\right)}^2+\mathrm{\pi }\times {\left(7\right)}^2\right]{\mathrm{cm}}^2 \\ \Rightarrow \mathrm{\pi }{r}^{\mathit{2}}=\left[\mathrm{\pi }\times 576+\mathrm{\pi }\times 49\right]{\mathrm{cm}}^2 \\ \Rightarrow \mathrm{\pi }{r}^{\mathit{2}}=\mathrm{\pi }\times \left(576+49\right){\mathrm{cm}}^2 \\ \Rightarrow r^{\mathit{2}}=625\mathrm{\pi }{\mathrm{cm}}^2 \\ \Rightarrow r^2=625 \\ \Rightarrow r=25 \end{gathered}$$

∴ Diameter of the new circle$=\left(25\times 2\right)\mathrm{cm}$
= 50 cm