A can in the shape of a right circular cylinder is required to have a volume of 250 cubic centimeters.

(a) Express the amount A of material to make the can as a function of the radius r of the cylinder.

(b) How much material is required if the can is of radius 3 centimeters?

(c) How much material is required if the can is of radius 5 centimeters?

(d) Graph $A=A\left(r\right)$. For what value of r is A smallest?

The given data is a can has a volume of 250 cubic centimeters.

Let us take radius of can as r cm and height of can as h cm.

We know volume of cylindrical can is ${\mathrm{\pi r}}^2\mathrm{h}$

Here we are given volume as 250 cubic centimeters. So, we get,

$$\begin{gathered} {\mathrm{\pi r}}^2\mathrm{h}=250 \\ \mathrm{h}=\frac{250}{{\mathrm{\pi r}}^2} \end{gathered}$$

The amount of material required to make the can is equal to the total surface area of the can.

$A=2{\mathrm{\pi r}}^2+2\mathrm{\pi rh}$

Substitute the value of height from above equation in the formula, we get,

$$\begin{gathered} A=2{\mathrm{\pi r}}^2+2\mathrm{\pi r}\left(\frac{250}{{\mathrm{\pi r}}^2}\right) \\ =2{\mathrm{\pi r}}^2+\frac{500}{\mathrm{r}} \end{gathered}$$

Substitute the value of radius as 3 in the formula obtained above to find the amount required.

$$\begin{gathered} A=2\left(\frac{22}{7}\right){\left(3\right)}^2+\frac{500}{3} \\ =56.52+166.66 \\ =223.19 \end{gathered}$$

Substitute the value of radius as 5 in the formula obtained above to find the amount required.

$$\begin{gathered} A=2\left(\frac{22}{7}\right){\left(5\right)}^2+\frac{500}{5} \\ =157+100 \\ =257 \end{gathered}$$

We need to plot the graph for A as a function of r. Thus, we get,

Therefore,$r=3.41cm$the amount of material required A would be smallest.