Question 924768
...perimeter a minimum?
{{{2x+2y=p}}};
{{{xy=400}}}
If "one of the sides" is x, then the other side is y. LOWER-CASE x; switching case level is incorrect unless you have a specific reason for using both case levels.


{{{y=400/x}}};
{{{2(400/x)+2x=p}}}
{{{highlight(p=800/x+2x)}}}


You want to know for what value of x will p be the minimum.


{{{p=800/x+2x*x/x}}}
{{{p=(800+2x^2)/x}}}
This seems not to help.  Going back to the red-outlined formula for p, the use of derivative should be useful, since I cannot right now think of a less advanced way.


{{{dp/dx=(-1)800x^(-2)+2}}}
{{{dp/dx=2-800/x^2}}}
Max or Min?
{{{highlight_green(2-800/x^2=0)}}}
{{{2=800/x^2}}}
{{{2*x^2=800}}}
{{{x^2=400}}}
{{{highlight(x=20)}}}


Is this x for a maximum or a minimum?
Letting the technology show us,
{{{graph(300,300,-4,35,-4,450,800/x+2x)}}}
Indicates that perimeter has a minumum.  The value there for the min perimeter should be {{{p=800/20+2*20}}}, {{{p=40+40=highlight(80)}}}.