can someone help me i dont think i fully understand the questionn and solving process?

Alan's answer is good.

Let's look at it this in a non-calculus manner.....

Let the parallel sides be x feet each......so, the remaining side = (120-2x)

So, the area is given by ...   x(120 - 2x)  = 120x - 2x^2 = -2x^2 + 120x

The x coordinate of the vertex of this parabola  = -b/2a (where -2 is a, and 120 is b) =  -120/2(-2) = 30

So this represents the length of one of the parallel sides.... and the remaining side is the longest = (120 - 2(30)) = 120 - 60 = 60 feet

And the max area occurs when x = 30......so we have  30(120 - 2(30)) = 30 * 60 = 1800 sq. ft.

The 120 feet of fencing will form 3 sides of the outdoor room.  Thw two sides touching the museum will have width W; the third side, joining these will have length L (which is what you are asked to find).

So we must have:  2W + L = 120  or  W = 60 - L/2

The enclosed area is given by A = W*L  or  A = (60 - L/2)*L  or 60*L - L²/2

To find the maximum area we differentiate A with respect to L and set the result to zero:

dA/dL = 60 - L

For dA/dL to be zero we must have L = 60

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