Question 310895
The area of a rectangle is,
{{{A=L*W=120000}}}
{{{L=120000/W}}}
The perimeter of a rectangle is,
{{{P=2*(L+W)}}}
Substituting,
{{{P=2*(120000/W+W)}}}
Now P is a function of W only.
To minimize P, find the derivative wrt W and set it equal to zero. 
{{{dP/dW=2*(-120000/W^2+1)=0}}}
{{{120000/W^2=1}}}
{{{W^2=120000}}}
{{{W=sqrt(120000)}}}
From the area equation above,
{{{L=120000/W=120000/sqrt(120000)=sqrt(120000)}}}
The rectangle with the smallest perimeter and area of 120000 sq. in. is a square with sides of {{{sqrt(120000)}}} in.