Optimization - Find the surface area of a domed cylinder?
"A metal storage tank with fixed volume V = 360pi m^3 is to be constructed from a right cylinder (including bottom) surmounted by a hemisphere. What dimensions will require the least amount of material?"
For surface area I used: 2(pi)rh + 3(pi)r^2 simplified to (pi)r(2h+3r)
To remove one of the variables I used: (pi)r(2(360/r^2)+3r)
The derivative is (6pi(r^3-120))/r^2
I'm not sure what to do after this however.
"A metal storage tank with fixed volume V = 360pi m^3 is to be constructed from a right cylinder (including bottom) surmounted by a hemisphere. What dimensions will require the least amount of material?"
For surface area I used: 2(pi)rh + 3(pi)r^2 simplified to (pi)r(2h+3r)
To remove one of the variables I used: (pi)r(2(360/r^2)+3r)
The derivative is (6pi(r^3-120))/r^2
I'm not sure what to do after this however.
1 Answer
Explanation:
The surface area
We know the volume is fixed at
This can be substituted into our formula for the SAO to eliminate
Finally, we can use the first derivative to find the critical value(s) of
We ignore the negative root since a negative radius would be meaningless in the context of the problem.
There are a few ways to verify this result is a minimum. I'll use the Second Derivative Test:
Since
The final part of the answer is