I have this exercise that I would like anyone to suggest the required steps in order to solve it
A cylindrical can is to be made to hold $250 \pi\; cm^3$. Find the dimensions of the can that will minimize the cost of the metal to manufacture it.
I have this exercise that I would like anyone to suggest the required steps in order to solve it
A cylindrical can is to be made to hold $250 \pi\; cm^3$. Find the dimensions of the can that will minimize the cost of the metal to manufacture it.
Volume is
$V=\pi r^2 h = 250\pi \implies h = \frac{250}{r^2}$
And thus, the surface area is given by
$A = 2\pi r^2+2\pi r h =\\ 2 \pi r^2 + 2\pi r\frac{250}{r^2} =\\ 2 \pi r^2 + 2\pi \frac{250}{r} =\\ 2 \pi r^2 + \frac{500 \pi}{r}$
Now minimize $A$.
Steps: