A right circular cone, a right circular cylinder and a hemisphere all have the same radius, and the heights of the cone and cylinder equal their diameters. Then their volumes are proportional respectively to:

Let the common radius of all three solids be $r$.
The height of the cone ($h_{\text{cone}}$) and the cylinder ($h_{\text{cylinder}}$) equals their diameters, so:

$$h_{\text{cone}}=h_{\text{cylinder}}=2r$$

Let us write down the volume formulas for each:

1. Volume of the cone ($V_1$):

$$V_1=\frac{1}{3}\pi r^2{h}_{\text{cone}}=\frac{1}{3}\pi r^2(2r)=\frac{2}{3}\pi r^3$$

2. Volume of the cylinder ($V_2$):

$$V_2=\pi r^2{h}_{\text{cylinder}}=\pi r^2(2r)=2\pi r^3$$

3. Volume of the hemisphere ($V_3$):

$$V_3=\frac{2}{3}\pi r^3$$

Now, compute the ratio $V_1:V_2:V_3$:

$$V_1:V_2:V_3=\frac{2}{3}\pi r^3:2\pi r^3:\frac{2}{3}\pi r^3$$

Divide out $\pi r^3$:

$$V_1:V_2:V_3=\frac{2}{3}:2:\frac{2}{3}$$

Multiply the entire ratio by $3$ to simplify:

$$V_1:V_2:V_3=2:6:2=1:3:1$$