A farmer wants to construct a pen with 100 feet of fencing. The pen will be divided into two areas as shown below. Find the maximum dimensions of the big pen.

Question

All I have so far is that 3x+2y=100. I am not sure how to move forward.

Answer 1

the area is

2xy = 2x(100-3x)/2 = 100x-3x^2

Now use what you know about parabolas to find the vertex (maximum area).

You will find that the total fencing is divided equally among the lengths and widths.

Answer 2

"as shown below" does not work here, but I have seen that problem many many times.

ok so far as your 3x + 2y = 100, assuming you defined y as the length and x as each of the dividing widths.
from that y = (100-3x)/2 = 50 - (3/2)x

I object to your wording of "Find the maximum dimensions of the big pen."
It should say:
"Find the dimensions of the big pen that will yield a maximum area."

Area = xy
= x(50-(3/2)x)
= 50x - (3/2)x^2

I will also assume that you are not in Calculus, or else this would now be easy

You are looking at a parabola which opens downwards. The vertex will give you all you want to know.
For a parabola in standard form, the x of the vertex is -b/(2a)
x = -50/(-3) = 50/3
then y = 50 - (3/2)(50/3) = 25

so the width is 25 ft and the length is 50/3 ft
for a maximum area of 1250/3 ft

If you have been finding the vertex by competing the square, I will leave it up to you to find the vertex of
(-3/2)x^2 + 50x using that method.