A specialized cooling tank for industrial lasers is being designed in the shape of an open-topped cuboid. The tank is constructed from high-grade sheet metal. The base of the tank has a length of 3x3x3x metres and a width of xxx metres. the vertical height of the tank is hhh metres.
Given that the volume of the tank must be exactly 162 m3162\text{ m}^3162 m3:
Show that the total surface area A m2A\text{ m}^2A m2 of the metal required is given by
A=3x2+432x A = 3x^2 + \frac{432}{x} A=3x2+x432Use calculus to find the value of xxx for which AAA is stationary, giving your answer to 3 significant figures.
Calculate the value of d2Adx2\frac{d^2A}{dx^2}dx2d2A at the stationary point and hence determine whether this value of xxx minimizes the amount of metal used.