Question 1033164
the volume of the cylinder with height {{{h}}} and radius {{{r}}} (both in cm) is
{{{pi*r^2*h=410}}} {{{cm^3}}} .
{{{pi*r^2*h=410}}} ---> {{{h=410/(pi*r^2)}}} .
The total surface area (including the top and bottom faces)of a right circular cylinder with height {{{h}}} and radius {{{r}}} is
{{{A=2pi*r*h+2pi*r^2}}} .
(We would measure {{{h}}} and radius {{{r}}} in cm, and {{{A}}} would be in {{{cm^2}}}, of course).
Substituting the expression found for {{{h}}} ,
{{{A=2pi*r*(410/(pi*r^2))+2pi*r^2}}} <--> {{{A=2*410/r+2pi*r^2}}} <--> {{{A=2*(410+pi*r^3)/r}}} .
 
We need to find the value of {{{r}}} that yields the minimum for {{{A}}} .
There may be another way to find that value.
Maybe you are expected to do it using a graphing calculator,
or maybe you are expected to use calculus,
and specifically derivatives.
The result should be the same.
Using calculus:
A local minimum of {{{A}}} happens only for a value of {{{r}}} that makes the derivative zero.
The derivative of {{{A=2*410/r+2pi*r^2}}} is
{{{dA/dr=-2*410/r^2+4pi*r=(2/r^2)(-410+2pi*r^3)}}} .
{{{(2/r^2)(-410+2pi*r^3)=0}}}<--->{{{-410+2pi*r^3=0}}}<--->{{{r^3=410/(2*pi)=205/pi}}} ---> {{{r=root(3,(205/pi))=highlight(4.03)}}} (correct to 2 decimal places).
Then, {{{h=410/(pi*r^2)=410/(pi*(205/pi)^(2/3))=2*(205/pi)^(2/3)=highlight(8.05)}}}