The given figure shows a hollow conical utensil and a hollow hemispherical utensil. The radius of each utensil is the same and when they are polished with the same kind of material, the cost of polishing them is also the same. What is the ratio of the depth of the hemispherical utensil to that of the conical utensil?

It is given that when the utensils are polished with the same kind of material, the cost incurred is the same. This means that their curved surface areas are the same.

Therefore,

C.S.A. of the conical utensil = C.S.A. of the hemispherical utensil

$\Rightarrow \pi rl=2\pi r^2$    [Given: Each utensil has the same radius.]

$\Rightarrow l=2r$

Depth of conical utensil $\left(h\right)=\sqrt{l^2-r^2}=\sqrt{{\left(2r\right)}^2-r^2}=r\sqrt{3}$ 

Also, the depth of the hemisphere is the same as its radius.

Thus, the required ratio

$=\frac{\mathrm{Depth} \mathrm{of} \mathrm{hemispherical} \mathrm{utensil}}{\mathrm{Depth} \mathrm{of} \mathrm{conical} \mathrm{utensil}}$

$=\frac{r}{r\sqrt{3}}$

$=\frac{1}{\sqrt{3}}$ 

$=1:\sqrt{3}$