If the farmer has 162 feet of fencing, what are the dimensions of the region which enclose the maximal area

The region is a square with a length side of 40.5 feet

Step-by-step explanation:

I will assume that the region is a rectangular area

we know that

The perimeter of the region is equal to

where

x is the length

y is the width

we have

so

simplify

----> equation A

The area of the rectangular region is

----> substitute equation A in equation B

This is the equation of a vertical parabola open downward

The vertex represent a maximum

The y-coordinate of the vertex represent the maximum area

The x-coordinate of the vertex represent the length for the maximum area

Convert the quadratic equation in vertex form

Factor -1

Complete the square

Rewrite as perfect squares

The vertex is the point (40.5,1,640.25)

so

The maximum area is

The length for the maximum area is

Find the value of y (equation A)

therefore

The region is a square with a length side of 40.5 feet

step-by-step explanation:

the answer is b