Question 1207300
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The volume of the cylindrical can with radius r and height h is to be 128pi cm^3:<br>
{{{V=(pi)r^2h=128pi}}} [1]<br>
The surface area of the can -- top, bottom, and side -- is<br>
{{{S=2(pi)r^2+2(pi)rh}}} [2]<br>
Solve [1] for h in terms of r and substitute in [2] to get an expression for the surface area in terms of the single variable r:<br>
{{{h=128/r^2}}}
{{{S=2(pi)r^2+2(pi)r(128/r^2)=2(pi)r^2+256(pi)/r}}}<br>
Find the derivative of the expression for the surface area and set it equal to zero to find the radius r that minimizes the surface area:<br>
{{{dS/dr=4(pi)r-256(pi)/r^2}}}
{{{4(pi)r-256(pi)/r^2=0}}}
{{{4(pi)(r-64/r^2)=0}}}
{{{r-64/r^2=0}}}
{{{r^3-64=0}}}
{{{r=4}}}<br>
The radius that minimizes the surface area is r=4; the corresponding height is 128/r^2 = 128/16 = 8.<br>
ANSWER: radius 4cm, height 8cm<br>