Answer:
The largest area is [tex]1,681\ un^2.[/tex] when the width is 41 units and the length is 41 units (when rectangle is a square)
Step-by-step explanation:
Let x units be the width of the rectangle, y units be the length of the rectangle.
If the perimeter of the rectangle is 164 units, then
[tex]2(x+y)=164\\ \\x+y=82\\ \\y=82-x\ units[/tex]
Find the area of the rectangle:
[tex]A=xy\\ \\A(x)=x(82-x)\\ \\A(x)=82x-x^2[/tex]
Find the derivative:
[tex]A'(x)=(82x-x^2)'=82-2x[/tex]
Equate it to 0:
[tex]82-2x=0\\ \\2x=82\\ \\x=41\ units[/tex]
When x = 41 units, the area is the largest and is equal to
[tex]A(41)=82\cdot 41-41^2=1,681\ un^2.[/tex]
