The Optimization of a Spherical Dome Tent

[Elainabeee]

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!

 

[Ray Vickson]

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!

When you write h2, do you mean $h^2$? If so, write it like this: h^2.

You have a constraint on (h,a) due to volume, and you must minimize the cost of materials, which will also be a function of the two variables h and a. Your expression for surface area does not account for the different costs per m^3 for the bottom and the top of the tent.

 

[Elainabeee]

When you write h2, do you mean $h^2$? If so, write it like this: h^2.

You have a constraint on (h,a) due to volume, and you must minimize the cost of materials, which will also be a function of the two variables h and a. Your expression for surface area does not account for the different costs per m^3 for the bottom and the top of the tent.

I meant h^2, sorry that was a typo! So if I want to minimize cost of materials, how can I solve for one of the variables, either h or a, in order to solve for the other? Or how can I eliminate one to later solve for the other?

 

[Ray Vickson]

Please give it a try yourself first, before asking for more help. Yes, I know, it is a struggle, but that is how you learn!

 

[Elainabeee]

Please give it a try yourself first, before asking for more help. Yes, I know, it is a struggle, but that is how you learn!

What I came up with when I tried is 2.2/1.6pi = h(3a^2+h^2)
from there, I can't get a variable alone

 

[Ray Vickson]

What I came up with when I tried is 2.2/1.6pi = h(3a^2+h^2)
from there, I can't get a variable alone

You don't have to; you can use the method of Lagrange Multipliers instead. See, eg., http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers.aspx ,
which goes through an example similar to yours in detail.

 

[Ray Vickson]

What I came up with when I tried is 2.2/1.6pi = h(3a^2+h^2)
from there, I can't get a variable alone

This equation is true if 1/6 = 1.6; otherwise, it is false. So, do you think it is true, or not?