Question 633890
It will be a 4 cm by 4 cm square.
 
Let the length of adjacent sides be {{{x}}} and {{{y}}}.
{{{drawing(200,150,0,20,0,15,
rectangle(1,2,19,13),
locate(9,1.9,x), locate(1.5,8,y)
)}}}
If {{{x=y}}}, you have a square, otherwise it is a rectangle.
The perimeter will be {{{x+y+x+y=2(x+y)=16}}} --> {{{x+y=8}}} --> {{{y=8-x}}}
The area will be {{{A=x*y=x(8-x)=8x-x^2}}}
We have area as a function of x.
We can write it as
{{{A(x)=-x^2+8x}}}
It is a quadratic function, that can be graphed as a parabola.
Like all quadratic functions
(in general written as {{{f(x)=ax^2+bx+c}}})
with a negative leading coefficient,
{{{A(x)}}} has a maximum at {{{x=-b/2a}}}.
In this case, that leading coefficient is {{{a=-1}}},
the number multiplying {{{x^2}}}.
The coefficient of the term in {{{x}}} is {{{b=8}}},
and the maximum is at {{{x=-8/2/(-1)}}} --> {{{highlight(x=4)}}}
The maximum occurs at