Perimeter of a rectangle=2(length)+2(width)
length=y
width=x
Then:
80=2x+2y
40=x+y
y=40-x
Then:
width=x
length=(40-x)
Area of a rectangle=length x width.
A(x)=x(40-x)
A(x)=40x-x²
We have to find the maximums or minimums of this function
1) we calculate the first derivative:
A´(x)=40-2x
2) we have to find the values of "x" when A´(x)=0
40-2x=0
-2x=-40
x=-40/-2
x=20
3) We have to calculate the second derivative.
A´´(x)=-2
Because A´´(x)<0; then we have a maximum at x=20
Therefore:
width: x=20
length: 40-x=40-20=20
This rectangle with maximum area is a square:
Area=(20 cm)(20 cm)=400 cm²
Answer: The lenght expressed in the terms of x would be: 40-x; and the maximun area would be 400 cm².
