TJC Inequalities Tutorial Q10

A wooden artwork is designed to take the form of a right pyramid with square base of sides \(2x\)m and fixed volume \(\frac{4}{3}\)m\(^{3}\). All its exterior faces, including the base, need to be painted. The cost per square metre of painting each of the \(4\) inclined faces is twice that of painting the base. Given that the cost of painting the base is \(\$p\)/m\(^{2}\), show that the total cost of painting the artwork is

\(\$\,4p{{x}^{2}}\left(1+2\sqrt{1+{{x}^{-\,6}}}\right)\).

Determine the possible heights the artwork may take if the budget given for the paint works is \(\$16p\).