(i)[5]
Write $h$ in terms of $x$ and hence prove that $A = \frac{3}{2}x^2 + \frac{24}{x}$.
(ii)[5]
When $x$ is allowed to vary, determine the value of $x$ for which $A$ is a minimum, making it clear that $A$ is minimised rather than maximised.
Mathematics 9709 · AS & A Level · Differentiation
Differentiation — practice question
The sketch depicts a rectangular tank of height $h$ metres with a lid fitted on top. Its base measures $x$ metres by $\frac{1}{2}x$ metres, and the lid is a rectangle with side lengths $\frac{5}{4}x$ metres and $\frac{4}{5}x$ metres. When the tank is completely filled, it contains $4\ \text{m}^3$ of water. The tank is made from material of negligible thickness. The total external surface area of the tank together with the area of the top of the lid is $A\ \text{m}^2$.