Answer:
The largest rectangle of perimeter 182 is a square of side 45.5
Step-by-step explanation:
Maximization Using Derivatives
The procedure consists in finding an appropriate function that depends on only one variable. Then, the first derivative of the function will be found, equated to 0 and find the maximum or minimum values.
Suppose we have a rectangle of dimensions x and y. The area of that rectangle is:
[tex]A=x.y[/tex]
And the perimeter is
[tex]P=2x+2y[/tex]
We know the perimeter is 182, thus
[tex]2x+2y=182[/tex]
Simplifying
[tex]x+y=91[/tex]
Solving for y
[tex]y=91-x[/tex]
The area is
[tex]A=x.(91-x)=91x-x^2[/tex]
Taking the derivative:
[tex]A'=91-2x[/tex]
Equating to 0
[tex]91-2x=0[/tex]
Solving
[tex]x=91/2=45.5[/tex]
Finding y
[tex]y=91-x=45.5[/tex]
The largest rectangle of perimeter 182 is a square of side 45.5
