If the volume of two spheres is in the ratio 27: 64, then the ratio of their radii is:

Given, the volume of two spheres = 27: 64

By formula, volume of a sphere = $\frac{4πr^3}{3}$, where r is the radius of the sphere.

Let r and R be the radii of the two spheres.

$$\begin{aligned}\Rightarrow \frac{\frac{4πr^3}{3}}{\frac{4πR^3}{3}} = \frac{27}{64} \\[1em] \Rightarrow \frac{r^3}{R^3} = \frac{27}{64} \\[1em] \Rightarrow \frac{r}{R} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} \\[1em] \Rightarrow \frac{r}{R} = \frac{3}{4}.\end{aligned}$$

Hence, option 1 is the correct option.