Question

Find the approximate radius and height of a can that holds 200cm^3 and uses the smallest amount of aluminum. Show clearly how you figured out the size of the can

Answer 1

\[V=\pi r^2h=200\] and
\[A=2\pi r h +2\pi r^2\] since the top and bottom of the can has area \(\pi r^2\) and the side of the can will have area of the circumference of the circle \(2\pi r\) times the height of the can \(h\)
solve the first equation for \(h\) and get \(h=\frac{200}{\pi r^2}\)
substitute in to the equation for the area and get
\[A=\frac{300}{r}+2\pi r^2\]
take the derivative, find the critical points etc
when you get done, the radius should be half the height, i.e. the diameter of the can should be equal to the height

Answer 2

Is there a simpler way to explain?

Answer 3

no