[It is given that the volume of a circular cone with base radius $r$ and height $h$ is \(\dfrac{

Question

[It is given that the volume of a circular cone with base radius $r$ and height $h$ is $\dfrac{1}{3}\pi r^2h$.]

The diagram above shows a cuboid with a square base with fixed sides $k$ cm and a fixed height $H$ cm inscribed in a right circular cone of base radius $r$ cm and height $h$ cm, with its base lying on the base of the cone.

Show that the volume of the cone, $V=\dfrac{\pi k^2h^3}{6(h-H)^2}$.[3]
Use differentiation to find, in terms of $H$, the value of $h$ for which $V$ is maximum. You do not need to show that $V$ is a maximum.[3]

Suppose $k=4$ and $H=10$. The dimensions of the cone vary such that its height $h$ increases at a constant rate of $2$ cm/s.

Find the rate of change of the volume of the cone at the instant when $h=15$.[3]

Accepted Answer

(a) $V=\dfrac{\pi k^2h^3}{6(h-H)^2}$ (b) No maximum exists; $h=3H$ gives a minimum (source error). (c) $-144\pi\text{ cm}^3\text{/s}$.

2026 HCI Promo Q10

2026 HCI Promo Q10

Solution:

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