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?
Now change the problem so that the floor is more expensive material than the rest; assume that the material is 1.4 times as expensive per square meter as the material making up the top of the tent.
(c)What should the dimensions of the ten be so that the cost of the material used is a minimum?
(d) what is the total area of the material used?

Steve · Answer

the area of a triangle of side s is √3/4 s^2

So, if the triangles have side x and the tent has length y, we have

(√3/4)x^2 * y = 2.2

the area of cloth used is

a = 2(√3/4)x^2 + 3xy

and the cost c is

c = 1.4(2(√3/4)x^2 + xy)+xy
Now just find dc/dx, after substituting in for y.

I assume that then you can do (b)-(d)