The ratio between the volumes of Sphere 1 and Sphere 2 is 27 : 64. Find the ratio of the surface areas of Sphere 1 and Sphere 2 respectively.

Let's assume the radius of Sphere 1 and Sphere 2 are 'R' and 'r' respectively.

The ratio between the volumes of Sphere 1 and Sphere 2 is 27 : 64.

$$\frac{\frac{4}{3}\times\pi\ \times R^3}{\frac{4}{3}\times\pi\ \times r^3}\ =\ \frac{27}{64}$$

$$\left(\frac{R}{r}\right)^3\ =\ \left(\frac{3}{4}\right)^3$$

Ratio of the surface areas of Sphere 1 and Sphere 2 = $$4\times\pi\times R^2\ \ :\ 4\times\pi\times r^2\ $$

= $$4\times\pi\times R^2\ \ :\ 4\times\pi\times r^2\ $$

Put Eq.(i) in the above equation.

= $$4\times\pi\times\left(3y\right)^2\ \ :\ 4\times\pi\times\left(4y\right)^2$$

= $$9y^2\ \ :\ 16y^2$$

= 9 : 16