- #1

Elainabeee

- 3

- 1

**1.**

If anyone can help me solve this optimization problem it would be much appreciated!

Imagine making a tent in the shape of a spherical cap (a sphere with a lower portion sliced away by a plane). Assume we want the volume to be 2.2m3. The floor of the tent is cheaper material than the rest: assume that the material making up the dome of the tent is 1.4 times as expensive per square meter than 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?

!

## Homework Equations

So far I have that the volume is 1/6pi*h (3a2+h2) with a being the radius of the base and h being the height. I also have surface area = 2*pi*r*h + pi(2rh-h2).

## The Attempt at a Solution

I know that you have to set the volume equation equal to 2.2 and solve for the dimensions. So I got

0= 1.6pi*h(3a^2+h^2) - 2.2

However, how would you solve having two variables? I can't figure out how to substitute into another equation to get just a or h in the volume equation.

Thank you so much for any help; if anyone can walk me through the steps I'd be so grateful!