If a cone and a cylinder have the same base area and the same volume, then what is the ratio of their respective heights?

Since the cone and cylinder have the same base area, their radii are the same.

Therefore, let radius of cone = radius of cylinder = r

Let h₁ and h₂ be the heights of the cone and the cylinder respectively.

∴ Volume of the cone = $\frac{1}{3}{\mathrm{\pi r}}^2{\mathrm{h}}_1$

Volume of the cylinder = πr²h₂

It is given that the volumes of the cone and the cylinder are equal.

$\therefore \frac{1}{3}{\mathrm{\pi r}}^2{\mathrm{h}}_1={\mathrm{\pi r}}^2{\mathrm{h}}_2$

⇒ h₁ = 3h₂

$\Rightarrow \frac{h_1}{h_2}=\frac{3}{1}$

Thus, the ratio of the heights of the cone and the cylinder is 3:1.