Question 858725
Find the volume of a cylinder, {{{V=pi*R^2*H=20*pi}}}
{{{R^2*H=20}}}
Find the total cost by calculating the area of the bottom and area of the side wall and multiplying by cost per unit area.
{{{T=pi*R^2(0.80)+2*pi*R*H*(0.32)}}}
{{{T=0.80*pi*R^2+0.64*pi*R*H}}}
From the volume equation, you can find a relationship between H and R.
{{{H=20/R^2}}}
Substitute,
{{{T=0.80*pi*R^2+0.64*pi*R*(20/R^2)}}}
{{{T=0.80*pi*R^2+(12.8*pi)*R^(-1)}}}
Now you have total cost T as a function of one variable.
Take the derivative and set it to zero.
{{{dT/dR=1.60*pi*R-12.8*pi*R^(-2)}}}
{{{1.60*pi*R-12.8*pi*R^(-2)=0}}}
{{{1.6*R=12.8.R^2}}}
{{{R^3=12.8/1.6}}}
{{{R^3=8}}}
{{{R=2}}}
Then,
{{{H=20/(2^2)=5}}}
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{{{T=0.80*pi*4+0.64*pi*2*5}}}
{{{T=10.053+20.106}}}
{{{T=30.16}}}