The diagram shows a solid object made from joining two hemispheres of radius $x$ cm to a cylinder of height $y$ cm. The volume of the object is 750 cm^3.

1. Formulate the equation for the total volume of the object

Volume of cylinder $= \pi x^2 y$
Volume of two hemispheres $= \frac{4}{3}\pi x^3$
Total volume $= \pi x^2 y + \frac{4}{3}\pi x^3 = 750$

2. Rearrange the equation to make $y$ the subject

$\pi x^2 y = 750 - \frac{4}{3}\pi x^3$
$y = \frac{750 - \frac{4}{3}\pi x^3}{\pi x^2}$
$y = \frac{750}{\pi x^2} - \frac{4}{3}x$

1. Express $y$ in terms of $x$ using the given volume (result from part a).

$\text{Volume} = \pi x^2 y + \frac{4}{3}\pi x^3 = 750$
$\pi x^2 y = 750 - \frac{4}{3}\pi x^3$
$y = \frac{750}{\pi x^2} - \frac{4}{3}x$

2. Formulate the total surface area $S$ in terms of $x$ and $y$.

$S = \text{Curved surface area of cylinder} + \text{Surface area of 2 hemispheres}$
$S = 2\pi x y + 4\pi x^2$

3. Substitute $y$ into the surface area equation and simplify to show the given result.

$S = 2\pi x \left( \frac{750}{\pi x^2} - \frac{4}{3}x \right) + 4\pi x^2$
$S = \frac{1500}{x} - \frac{8}{3}\pi x^2 + 4\pi x^2$
$S = \frac{4}{3}\pi x^2 + \frac{1500}{x}$

1. Differentiate $S$ with respect to $x$

$S = \frac{4}{3}\pi x^2 + 1500x^{-1}$

$\frac{\text{d}S}{\text{d}x} = \frac{8}{3}\pi x - 1500x^{-2}$

2. Set the derivative to zero to find the stationary point

For a minimum value, $\frac{\text{d}S}{\text{d}x} = 0$:

$\frac{8}{3}\pi x - \frac{1500}{x^2} = 0$

$\frac{8}{3}\pi x = \frac{1500}{x^2}$

3. Solve for $x$

$x^3 = \frac{1500 \times 3}{8\pi}$

$x^3 = \frac{1125}{2\pi}$

$x = \sqrt[3]{\frac{1125}{2\pi}} \approx 5.63625$

4. Calculate the minimum value of $S$

Substitute $x = 5.63625$ into $S$:

$S = \frac{4}{3}\pi (5.63625)^2 + \frac{1500}{5.63625}$

$S \approx 132.938 + 266.134 = 399.072$

$S \approx 399$ (to 3 s.f.)