Question 310541
Perimeter of a rectangle
{{{P=2*(L+W)}}}
Area of a rectangle
{{{A=L*W=500}}}
{{{L=500/W}}}
Substitute this expression into the perimeter equation,
P=2*(500/W+W)
Now perimeter is a function only of W.
To find the minimum, take the derivative wrt W and set it equal to zero.
{{{P=1000/W+2W=1000W^(-1)+2W}}}
{{{dP/dW=-1000W^(-2)+2=2-1000/W^2=0}}}
{{{1000/W^2=2}}}
{{{W^2=500}}}
{{{W=sqrt(500)}}}
Since {{{L*W=500}}}, then
{{{L=sqrt(500)}}}
The shape that minimizes perimeter is a square of side {{{sqrt(500)}}}.
{{{Pmin=4sqrt(500)}}}