[It is given that the surface area and the volume of a sphere of radius $r$ is $4\pi r^2$ and $\frac{4}{3}\pi r^3$ respectively.] A manufacturer produces barriers which consist of a hemisphere of radius $r\text{ cm}$ joined to a cylinder of base radius $r\text{ cm}$ and height $h\text{ cm}$ (see diagram). The manufacturer decides that the external surface area, excluding the base, of each barrier is $16200\text{ cm}^2$.
Using differentiation, find the exact value of the maximum volume of a barrier.

1. Formulate an equation for the external surface area excluding the base.

The external surface area consists of the curved surface area of the cylinder and the curved surface area of the hemisphere.
$2\pi rh + 2\pi r^2 = 16200$

2. Express the height $h$ in terms of $r$.

$2\pi rh = 16200 - 2\pi r^2$
$h = \frac{16200 - 2\pi r^2}{2\pi r}$
$h = \frac{8100}{\pi r} - r$

3. Formulate an equation for the volume of the barrier in terms of $r$ and $h$.

The volume $V$ is the sum of the volumes of the cylinder and the hemisphere:
$V = \pi r^2 h + \frac{2}{3}\pi r^3$

4. Express the volume $V$ entirely in terms of $r$.

$V = \pi r^2 \left( \frac{8100}{\pi r} - r \right) + \frac{2}{3}\pi r^3$
$V = 8100r - \pi r^3 + \frac{2}{3}\pi r^3$
$V = 8100r - \frac{1}{3}\pi r^3$

5. Differentiate $V$ with respect to $r$.

$\frac{\mathrm{d}V}{\mathrm{d}r} = 8100 - \pi r^2$

6. Equate the derivative to zero to find the stationary value of $r$.

For maximum volume, $\frac{\mathrm{d}V}{\mathrm{d}r} = 0$:
$8100 - \pi r^2 = 0$
$\pi r^2 = 8100$
$r^2 = \frac{8100}{\pi}$
$r = \frac{90}{\sqrt{\pi}} \quad \text{(since } r > 0 \text{)}$

7. Verify that this value of $r$ gives a maximum volume using the second derivative test.

$\frac{\mathrm{d}^2V}{\mathrm{d}r^2} = -2\pi r$
When $r = \frac{90}{\sqrt{\pi}}$, $\frac{\mathrm{d}^2V}{\mathrm{d}r^2} = -2\pi \left( \frac{90}{\sqrt{\pi}} \right) < 0$.
Since the second derivative is negative, the volume is maximized.

8. Calculate the exact maximum volume.

$V = 8100 \left( \frac{90}{\sqrt{\pi}} \right) - \frac{1}{3}\pi \left( \frac{90}{\sqrt{\pi}} \right)^3$
$V = \frac{729000}{\sqrt{\pi}} - \frac{1}{3}\pi \left( \frac{729000}{\pi\sqrt{\pi}} \right)$
$V = \frac{729000}{\sqrt{\pi}} - \frac{243000}{\sqrt{\pi}}$
$V = \frac{486000}{\sqrt{\pi}}$