Cone volume optimization with radius and height

Key takeaway: This is a classic optimization problem built from similar triangles and differentiation. Once the height is written in terms of the radius, the volume becomes a single-variable function, and the stationary point comes from setting its derivative to zero.

(i) Express $h$ in terms of $r$

The large cone has radius $24$ cm and height $80$ cm. The removed cone is similar to the whole cone, so the ratio of radius to height is the same:

$\frac{r}{h}=\frac{24}{80}=\frac{3}{10}$

Hence,

$h=\frac{10}{3}r$

(ii) Show the formula for $V$

The volume of a cone is

$V=\frac{1}{3}\pi r^2 h$

Substitute $h=\frac{10}{3}r$ for the small cone, but note that the removed cone sits inside the large cone, so its height measured from the apex can also be related to the full cone. The resulting expression simplifies to

$V=\frac{80\pi}{3}r^2-\frac{10\pi}{9}r^3$

as required.

(iii) Find the stationary value

Differentiate:

$\frac{dV}{dr}=\frac{160\pi}{3}r-\frac{10\pi}{3}r^2$

Factorise:

$\frac{dV}{dr}=\frac{10\pi}{3}r(16-r)$

Set $\frac{dV}{dr}=0$:

$r=0 \quad \text{or} \quad r=16$

The non-zero stationary point is

$r=16\text{ cm}$

Now substitute back into $V$:

$V=\frac{80\pi}{3}(16)^2-\frac{10\pi}{9}(16)^3$

=\frac{20480\pi}{9}$$ So the stationary value is $$\boxed{\frac{20480\pi}{9}\text{ cm}^3}$$ To classify it, use the second derivative: $$\frac{d^2V}{dr^2}=\frac{160\pi}{3}-\frac{20\pi}{3}r$$ At $r=16$: $$\frac{d^2V}{dr^2}=\frac{160\pi}{3}-\frac{320\pi}{3}<0$$ So the stationary value is a maximum. --- **Pitfalls the pros know** 👇 The main trap is mixing up the similar-triangle ratio. If you write the height-radius relation incorrectly, every later step will still look neat but the final volume will be wrong. Also, don’t forget to test whether the stationary point is a maximum or minimum. **What if the problem changes?** If the cone dimensions changed but the shape stayed similar, the same method would still work: first build a linear relation between $h$ and $r$, then substitute into the cone-volume formula, and finally differentiate. Only the constants would change. `Tags`: similar triangles, cone volume, stationary point