A container consists of a right cylindrical part and a hemispherical part at the top, as shown in the picture and diagram below - NSC Technical Mathematics - Question 8 - 2019 - Paper 1

To find the total volume of the container, we sum the volumes of the cylindrical part and the hemispherical part. The volume of the cylindrical part is:

$V_{cylinder} = \pi x^2 h = \pi x^2 (66 - 3x) = 66\pi x^2 - 3\pi x^3$

The volume of the hemispherical part is:

$V_{hemisphere} = \frac{2}{3} \pi x^3$

Thus, the total volume $V$ is:

$V = V_{cylinder} + V_{hemisphere} = (66\pi x^2 - 3\pi x^3) + \frac{2}{3}\pi x^3 = 66\pi x^2 - \left(3 - \frac{2}{3}\right)\pi x^3 = 66\pi x^2 - \frac{7}{3}\pi x^3$

To maximise the volume, we first find the derivative of the volume function:

$V(x) = 66\pi x^2 - \frac{7}{3}\pi x^3$

Taking the derivative:

$\frac{dV}{dx} = 132\pi x - 7\pi x^2$

Setting the derivative equal to zero to find critical points:

$0 = 132\pi x - 7\pi x^2$

Factoring out common terms:

$\pi x (132 - 7x) = 0$

This yields:

$x = 0 \text{ or } x = \frac{132}{7}$

The value $x = 0$ is not valid in this context, so we check:

$x = \frac{132}{7}$

Substituting $x = \frac{132}{7}$ back into the volume formula:

$V\left(\frac{132}{7}\right) = 66\pi \left(\frac{132}{7}\right)^2 - \frac{7}{3}\pi \left(\frac{132}{7}\right)^3$

Calculating each term:

$= 66\pi \frac{17424}{49} - \frac{7}{3}\pi \frac{2299968}{343}$

Obtaining a common denominator and simplifying gives:

$\approx 24,576.74 \text{ or } 7823.02\pi$

Thus, the maximum volume of the container is approximately $24,576.74\text{ cm}^3$.