my brain hurts me and i have to submit tomorrow 😭

[exotic hull]

Suppose the radius and height of a can are r and h. Then its surface area is:
S = 2πr(r + h)
And its volume is:
V = πr^2 h
As S is constant, letcs express h from it.
r + h = S/(2πr)
h = S/(2πr) - r
Now we substitute it into the extression for V:
V = πr^2 (S/(2πr) - r) = (S/2)r - πr^3
So, now you can find the maximum point of this function. Before that, we need to find an interval for which the dimensions make sense - they must both be positive:
r > 0
h = S/(2πr) - r > 0
Thus, we get 0 < r < √(S/(2π)).
So, find the maximum value of the folloeing function V(r) for our interval that we found:
V(r) = (S/2)r - πr^3, 0 < r < √(S/(2π))
After you do that, find h = S/(2πr) - r and you're done.

I highly recommend only substituting S at the VERY end - when you already obtain expressions for r and h.

[thorn bay]

hi