Question 975337
A cylindrical can is to hold 16πcm3 of mango juice. The cost per square meter of constructing the top and bottom is twice the cost per square meter of constructing the cardboard side. What dimensions minimize the total cost of construction?
===============
Area of top + bottom = {{{2pi*r^2}}}
Lateral area = {{{2pi*r*h}}}
{{{Vol = pi*r^2h = 16pi}}}
---------------
{{{Cost = k*(2pi*r*h + 4pir^2)}}} where k is some constant
----
{{{Vol = pi*r^2h = 16pi}}}
{{{r^2h = 16}}}
{{{h = 16/r^2}}}
Sub for h in the cost eqn:
{{{C = k*(32pi/r + 4pir^2)}}}
Find the 1st derivative of Cost wrt r, set = 0:
dC/dr = {{{k*(-32pi/r^2 + 8pi*r) = 0}}}
{{{-32/r^2 + 8r = 0}}}
{{{r - 4/r^2 = 0}}}
{{{r^3 - 4 = 0}}}
{{{r = root(3,4)}}}