The ratio of heights of $a$ cone and $a$ hemisphere with equal bases and equal volumes is ________.

Step 1 (Setup volume equations): Let the base radius of both the cone and the hemisphere be $r$, since they have equal bases.

• Volume of a cone: $V_1=\frac{1}{3}\pi r^2{h}_{\text{cone}}$

• Volume of a hemisphere: $V_2=\frac{2}{3}\pi r^3$

Step 2 (Equate the volumes): We are given that they have equal volumes:

Dividing both sides by $\frac{1}{3}\pi r^2$:

Step 3 (Ratio of Heights): The height of the hemisphere is equal to its radius $r$. The ratio of the height of the cone to the height of the hemisphere is:

Step 4 (Conclusion): The ratio of the heights is exactly 2 (or 2:1).