A cake in the shape of a right circular cone of height h and base radius r is to be cut parallel to the base. At what distance from the top should the cake be cut to get two parts of equal volumes?

Correct option is A

Given:
A cake is in the shape of a right circular cone with:
Height = h
Base radius = r
We are to cut the cone parallel to the base such that the top and bottom parts have equal volume.
We are to find the distance from the top where the cut is made.

Concept Used:
Volume of a cone = (1/3)πr²h
If a cone is cut parallel to the base, the smaller cone on top and the frustum at the bottom are similar in shape.
Let the cut be made at height x from the top, then:
The smaller cone formed at the top has height x
Its radius is (r/h)·x because of similarity
Volume of small (top) cone:

​$$\\[1em]\textbf{Volume of the small (top) cone:} \\V_{\text{small}} = \frac{1}{3} \pi \left( \frac{r}{h}x \right)^2 x = \frac{1}{3} \pi \cdot \frac{r^2 x^3}{h^2}\\[1em]\textbf{Volume of the full cone:} \\V_{\text{full}} = \frac{1}{3} \pi r^2 h\\[1em]\textbf{Since both parts have equal volume:} \\V_{\text{small}} = \frac{1}{2} V_{\text{full}} \\\ \\\Rightarrow \frac{1}{3} \pi \cdot \frac{r^2 x^3}{h^2} = \frac{1}{2} \cdot \frac{1}{3} \pi r^2 h\\[1em]\text{Cancel } \frac{1}{3} \pi r^2 \text{ from both sides:} \\\frac{x^3}{h^2} = \frac{h}{2} \\\ \\x^3 = \frac{h^3}{2} \\\ \\x = \frac{h}{2^{1/3}}$$

Correct Answer: Option A​​