Two cylinders have their heights in the ratio 1 : 2 and their radii in the ratio 2 : 1. What is the ratio of their volumes?

Two cylinders have their heights in the ratio 1 : 2 and their radii in the ratio 2 : 1.

Let's assume the radius of two cylinders $$R_1$$ and $$R_2$$

So assume $$R_1 = 2y$$ and $$R_2 = y$$.

Let's assume the height of two cylinders $$H_1$$ and $$H_2$$

So assume $$H_1 = z$$ and $$H_2 = 2z$$.

$$\frac{volume\ of\ first\ cylinder}{volume\ of\ \sec ond\ cylinder}\ =\frac{\pi\ \times\ \left(R_1\right)^2\ \times\ H_1}{\pi\ \times\ \left(R_2\right)^2\ \times\ H_2}$$

= $$\frac{\pi\ \times\ \left(2y\right)^2\ \times z}{\pi\ \times\ \left(y\right)^2\ \times2z}$$

= $$\frac{4y^2\ \times z}{y^2\ \times2z}$$

= $$\frac{2}{1}$$

So the ratio of their volumes is 2:1.