Question 1142711
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As you know, the volume of a cylinder is 

    V = {{{pi*r^2*h}}}, 

where pi = 3.14, r is the radius and h is the height.


In your case the volume is fixed:

    {{{pi*r^2*h}}} = 1000 cubic centimeters.                   (1)


The surface area of a cylinder is 

    S = {{{2pi*r*h}}} + {{{2pi*r^2}}},                               (2)

and they ask you to find minimum of (2) under the restriction (1).


You can rewrite the formula (2) in the form

    S(r) = {{{(2*pi*r^2*h)/r}}} + {{{2pi*r^2}}}.                         (3)


In formula (3), replace  {{{pi*r^2*h}}}  by  1000, based on (1). You will get

    S(r) = {{{(2*1000)/r}}} + {{{2pi*r^2}}} = {{{2000/r}}} + {{{2pi*r^2}}}.


The plot below shows the function S(r) = {{{2000/r}}} + {{{2pi*r^2}}}, and you can clearly see that it has the minimum.



    {{{graph( 330, 330, -5, 20, -50, 1000,
          2000/x + 2*3.14*x^2
)}}}


        Plot y = {{{2000/r}}} + {{{2*3.14*r^2}}}



To find the minimum, use Calculus: differentiate the function to get

S'(r) = {{{-2000/r^2}}} + {{{4pi*r}}} = {{{(-2000 + 4pi*r^3)/r^2}}}

and equate it to zero.


S'(r) = 0   leads you to equation  {{{4pi*r^3}}} = {{{2000}}},   which gives 

r = {{{root(3,500/pi)}}} = {{{root(3,500/3.14)}}} = 5.42 cm (approximately).


<U>Answer</U>.  r = 5.42 cm, h = {{{1000/(3.14*5.42^2)}}} = 10.84 cm  give the minimum of the surface area.
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