There is a farmer that wants to enclose a pasture for his cows. He has 600 yards of fencing for the perimeter of the field. What are the dimensions of the largest square field he can​ enclose?

Question

Given the P=600 and P=2L+2W I would think the answer is 150yd by 150yd, is this correct?

Answer 1

Perimeter = 2l+ 2w

2l + 2w = 600

Area = l* w

We need to solve the perimeter formula for either l or w. Let’s solve for w:

 

2l + 2w = 600

2w = 600- 2l

w =( 600-2l)/ 2

w = 300 - l

Now substituting w= 300-i  into the area formula we have:

Area = l*w

Area = l * (300-i)

Area = 300l -l^2

A =300l -l^2

Since A represents a quadratic equation  () in terms of l, we will re-write A in function form with the exponents in descending order:

A(l) =-l^2 +300l

As equation represent Parabola and a is negative parabola have it's maximum point at it vertex.

The y-coordinate of the vertex will represent maximum area. we need to find the value of the x-coordinate of the vertex. (that is, the value of l in our equation).

l= -b/2a

l= -300/2(-1)

l=300/2 = 150 yards

Substituting this value for l into our equation for area yields:

a(l) = -l ^2 +300l

a(150) =-(150)^2 +300(150)

=- 22500 + 45000

  =22500

So the dimensions of the largest square field he can​ enclose is = 22500 Square yards