Imagine making a tent in the shape of a right prism whose cross-section is an equilateral triangle (the door is on one of the triangular ends). Assume we want the volume to be 2.2 m3, to sleep two or three people. The floor of the tent is cheaper material than the rest: assume that the material making up the ends and the top of the tent is 1.4 times as expensive per square meter as the material touching the ground.

Question

Imagine making a tent in the shape of a right prism whose cross-section is an equilateral triangle (the door is on one of the triangular ends). Assume we want the volume to be 2.2 m3, to sleep two or three people. The floor of the tent is cheaper material than the rest: assume that the material making up the ends and the top of the tent is 1.4 times as expensive per square meter as the material touching the ground.
(a) What should the dimensions of the tent be so that the cost of the material used is a minimum?
(b) What is the total area of the material used?

Steve · Answer

If the tent ends have side s and the tent has length t, the area is

a = 2(√3/4 s^2) + 3st
since the volume of the tent is 2.2,
√3/4 s^2 t = 2.2, and so
t = 8.8/(√3 s^2)

If the cost of the floor is 1, then the cost of the whole tent is

c = 2(√3/4 s^2)(1.4) + 2st + 1.4st
= 0.7√3 s^2 + 3.4s(8.8/(√3 s^2))
= 1.212 s^2 + 17.274/s
dc/ds = 2.414s - 17.274/s^2
= (2.414s^3 - 17.274)/s^2
dc/ds=0 when s=1.927
so, t=1.368