Solving the Pup Tent Problem: Volume & Cost Minimization

[Todd Bakker]

"The pup tent problem"

This has been assigned for our introduction to calculus class and we are completely stuck, any ideas on how to solve this problem? Thanks for the help!

The Dome Tent
Imagine making a tent in the shape of a spherical cap (a sphere with lower portion sliced away by a plane). Assume we want the volume to be 2.2 m^3, to sleep two or three people.
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
What should the dimensions of the tent be so that the cost of the material is a minimum?

Also, what formulas should we use for Surface area and Volume(we have found many different ones online)
Thanks so much for your time and brain power!

 

[dynamicsolo]

The Dome Tent
Imagine making a tent in the shape of a spherical cap (a sphere with lower portion sliced away by a plane). Assume we want the volume to be 2.2 m^3, to sleep two or three people.
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
What should the dimensions of the tent be so that the cost of the material is a minimum?

Also, what formulas should we use for Surface area and Volume(we have found many different ones online)

You'll have to show some sort of attempt to work out the problem in order to get help here.

I assume that since the problem specificies a "spherical cap" and not a hemisphere, you will need to look online for the formulas for the surface area and volume of a spherical cap of height h and sphere radius R. Don't forget that the materials for the tent will include a circular base made of different material from the dome. (If you're in a second-semester calculus course, you may be expected to derive the formulas for yourself.)