1.22: A solid cuboid has width cm, length cm and height cm.

A solid cuboid has width $\displaystyle x$ cm, length $\displaystyle 4x$ cm and height $\displaystyle h$ cm.
The volume of the cuboid is $\displaystyle 75 \mathrm{cm}^3$ and the surface area of the cuboid is $\displaystyle S \mathrm{cm}^2$

(a) Show that $\displaystyle S = 8x^2 + \frac{375}{2x}$

Given that $\displaystyle x$ can vary, using calculus,

(b) (i) find to 3 significant figures, the value of $\displaystyle x$ for which $\displaystyle S$ is a minimum,

(ii) justify that this value of $\displaystyle x$ gives a minimum value of $\displaystyle S$

(c) Find, to 3 significant figures, the minimum value of $\displaystyle S$