Question 939772
If you need to design a cylinder with a volume of {{{"50,000"}}}{{{cm^3}}} ,
you could start with
{{{R}}}= radius of the bases of the cylinder, and
{{{H}}}= height of the cylinder,
both in {{{cm}}} .
The volume in {{{cm^3}}} is calculated as
{{{pi*R^2*H}}} .
If that volume is {{{"50,000"}}}{{{cm^3}}} , then
{{{pi*R^2*H="50,000"}}}-->{{{H="50,000"/(pi*R^2)}}} .
Now, if you make it a right cylinder
(one with its axis perpendicular to the bases,
the total surface area is calculated as
{{{A=2pi*R^2+2pi*R*H}}} .
Substituting {{{H="50,000"/(pi*R^2)}}} into the equation above, we get
{{{A=2pi*R^2+2pi*R*("50,000"/(pi*R^2))}}}
{{{A=2pi*R^2+2*"50,000"/(pi*R))}}}
{{{A=2pi*R^2+"100,000"/(pi*R))}}}
The area as a function of radius of the cylinder
is very large for
very narrow (and very tall) cylinders, and for
very short and very wide) cylinders with the same area
Its graph looks like this
{{{graph(300,300,-10,40,-1000,9000,2pi*x^2+100000/(pi*x))}}}
In the middle, there is a value for the radius that makes the surface are as small as possible.
How would you find that value for the radius, and the minimum area?
Maybe you are expected to use a graphing calculator.
Maybe you are studying calculus.
and are expected to calculate the derivative of the function
{{{A=2pi*R^2+"100,000"/(pi*R))}}} ,
and find the {{{R}}} value where that derivative is zero ans {{{A}}} is minimum.
The derivative is
{{{dA/dR=4pi*R-100000/(pi*R^2)}}}<-->{{{dA/dR=(4pi^2*R^3-100000)/(pi*R^2)}}}
and that is zero when
{{{4pi^2*R^3=100000}}}-->{{{R^3=100000/4pi^2}}}-->{{{R=root(3,100000/4pi^2)}}}-->{{{R=13.63(rounded)}}} ,
and the approximate value for the minimum area (in cubic cm)
(when {{{R}}} takes that value) is 
{{{A=3502.6}}}