Question

A tent of a given volume has a square base of side 2a, has its four sides vertical of height b and is surmounted by a regular pyramid of height 1. find the values of a and b in terms of height I such that the canvas required for its construction is minimum.

Answer 1

area of canvas:

$S=4a\sqrt{h^2+a^2}-2ah$

$S'_h=\frac{4ha^2}{\sqrt{a^2h^2+a^4}}-2a=0$

$4h^2=h^2+a^2$

$a=h\sqrt 3$

$b^2=a^2+h^2=4h^2$

$b=2h$