Question

Optimization problem: what are the dimensions of the lightest open-top right circularcylindrical can that will hold a volume of 1000cm3?Compare the result with the result in example 2.

Answer 1

Do we assume that the weight of the can is proportional to the surface area of the can, which assuming constant thickness is a measure of the amount of material?

Answer 2

I'm gonna assume so, anyways. I'm also going to assume that you know / are learning about Lagrange multipliers, because this question begs for them. The function being minimized is the surface area,
\[ S(r,h) = \pi r^2 + 2\pi r h\]
where r is the radius of the cylinder and h is it's height. The constraint is that the volume is 1000, which we can write as
\[ F(r,h) = \pi r^2 h - 1000 = 0\]
We now say that
\[\vec{\nabla S}= \lambda \vec{\nabla}F \]
or
\[ \left<2\pi(r+h), 2\pi r \right> = \lambda\left< 2\pi r h , \pi r^2\right>\]
which yields
\[2\pi(r+h) = \lambda \cdot 2\pi r h\]
\[2\pi r = \lambda \cdot \pi r^2 \]
combined, this yields
So
\[ \lambda = \frac{2}{r} \]
plugging that in,
\[2\pi (r+h) = 4\pi h\]
yielding
\[r+h = 2h\]
so r = h. We can determine what they are using our constraint:
\[V = \pi r^2 h = \pi r^3 = 1000 \rightarrow r = h \approx 6.83 \space \text{cm} \]

Answer 3

This is about finding the max or min, so this is not exactly the way it wants me to do. But, it seems like i was using the wrong formula for the surface area, which we have to minimize. anyways thank you Jemurray3.

Answer 4

Purely for my enlightenment, what are you trying to do then? The technique I just used minimizes or maximizes a function subject to some constraint. The function was surface area, the constraint was its volume. What is the result from "example 2"?

Answer 5

it's just those notations that confused me sorry about that.