Question 484460
A clyinderical can of radius xcm has volume 144cm^3. the cost of producing the can is determined by it's surface area.

(A) show that the height of the can is h=144/piex^2.

(B) find an expression for the surface area of the can

(C) find the dimensions that will minimise the cost of production.
=============================================================================
The volume of the can is V = {{{pi*x^2*h}}} where h = the height of the can
A) Solve for h in the above equation and substitute the value given for V:
{{{h = 144/(pi*x^2)}}}
B) The surface area of the can is given by {{{S = 2*pi*x*h + 2*pi*x^2}}}
Substitute the expression for h derived in A) above:
{{{S = 2*pi*x*144/(pi*x^2) + 2*pi*x^2}}} 
{{{S = 288/x + 2*pi*x^2}}}
C) The surface area will be a minimum where dS/dx = 0
{{{dS/dx = 0 = -288/x^2 + 4*pi*x}}}
Solve for x:
Multiply through by {{{x^2}}}
{{{4*pi*x^3 = 288}}}
{{{x = (72/pi)^(1/3)}}}
The value for h can then be obtained from the expression derived in A).