The diameters of two solid spheres are in the ratio 5 : 7. The ratio between areas of their curved surfaces is :

1. 5 : 7

2. 7 : 5

3. 49 : 25

4. 25 : 49

Given,

The diameters of two solid spheres are in the ratio 5 : 7.

Let the diameters of two spheres be 5a and 7a.

So, their radius will be r₁ = $\dfrac{5a}{2}$ and r₂ = $\dfrac{7a}{2}$

By formula,

Curved surface area of sphere = 4πr²

$\text{Ratio of their curved surfaces areas }= \dfrac{4πr$1^2}{4πr2^2}\\[1em] = \dfrac{4 \times π \times \Big(\dfrac{5a}{2}\Big)^2}{4 \times π \times \Big(\dfrac{7a}{2}\Big)^2}\\[1em] = \dfrac{\Big(\dfrac{5a}{2}\Big)^2}{\Big(\dfrac{7a}{2}\Big)^2}\\[1em] = \dfrac{\Big(\dfrac{25a^2}{4}\Big)}{\Big(\dfrac{49a^2}{4}\Big)}\\[1em] = \dfrac{25}{49}\\[1em] = 25 : 49.

Hence, option 4 is the correct option.