Question

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.

Answer 1

`r = 12sqrt(3), h = 5/6-8sqrt(3)`

Explanation:

The surface area `S` for the tank would be comprised of the surface area of the body of the cylinder + area of the circle base + area of the hemisphere on the top. Recall that:

`SA = underbrace(2pirh)_ "cyl" + underbrace(pir^2)_ "base" + underbrace(2pir^2)_ "hemi" = 2pirh + 3pir^2`

We know the volume is fixed at `360pi`, which means we can use the volume of the tank to isolate a variable and remove it from the SA formula:

`V = underbrace(pir^2h)_ "cyl" + underbrace(2/3pir^3)_ "hemi" = pir^2(h+2/3r)`

`pir^2(h+2/3r) = 360pi`

`r^2(h+2/3r) = 360`

`h + 2/3r = 360/r^2`

`h = 360/r^2 - 2/3r`

This can be substituted into our formula for the SAO to eliminate `h` from the formula:

`SA = 2pih+3pir^2 = 2pir(360/r^2-2/3r)+3pir^2`

`= (720pi)/r-(4pi)/3r^2+3pir^2 = (720pi)/r+(5pi)/3r`

Finally, we can use the first derivative to find the critical value(s) of `r`, and verify that they represent a minimum as requested.

`SA' = 0`

`-(720pi)/r^2+(5pi)/3 = 0`

`(5pi)/3 = (720pi)/r^2`

`r^2 = (720pi*3)/(5pi) = 432`

`r = sqrt(432) = 12sqrt(3)`

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:

`SA'' = (1440pi)/r^3 > 0 " for " r = 12sqrt(3)`

Since `SA''>0`, the value we found represents a minimum.

The final part of the answer is `h`:

`h = 360/r^2 - 2/3r = 360/(sqrt(432))^2 - 2/3(12sqrt(3))`

`h = 360/432 - 8sqrt(3) = 5/6 - 8sqrt(3)`