14. The diagram below shows a closed box in the form of a cuboid, which is such that the length of its base is twice the width of its base. The volume of the box is \(9000 \mathrm {~cm} ^ { 3 }\). The total surface area of the box is denoted by \(S \mathrm {~cm} ^ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-5_357_915_1190_543}
- Show that \(S = 4 x ^ { 2 } + \frac { 27000 } { x }\), where \(x \mathrm {~cm}\) denotes the width of the base.
- Find the minimum value of \(S\), showing that the value you have found is a minimum value.