
minimize problem
The consultancy unit of a university is to introduce a new soft drink to be packed in aluminium cans each to hold 45cl.find the dimensions that will minimize the amount of material used in the construction of the can ,assuming the thickness of the material is uniform.

Since the shape of a can is a cylinder, let $\displaystyle r$ be the radius of the base and let $\displaystyle h$ be the height of the cylinder.
From given we have $\displaystyle V=\pi r^2 h=45$, further let $\displaystyle S$ be the surface area of the cylinder. And $\displaystyle S= 2\pi r^2 + 2\pi r h$. Our goal is to minimize the surface area.
From the given volume constraint, let's rewrite $\displaystyle h=\frac{45}{\pi r^2}$, plug into the surface formula above, we have:
$\displaystyle S=2\pi r^2 + \frac{90}{r}$
In order to find the minimum value of $\displaystyle S$, let's find the critical number(s) of the surface function by:
$\displaystyle \frac{d}{dr}S = 4\pi r90r^{2}=0$, which gives $\displaystyle r=\sqrt[3]{\frac{90}{4\pi}}\approx 1.9276$ (please verify it is indeed a minimum)
Then $\displaystyle h=\frac{45}{\pi r^2} \approx 3.8552 $