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Regarding this post, I have two notices.
1. Your formulation is INCORRECT.
The question should ask about the MINIMUM length of the fencing ---- NOT about the maximum length.
The maximum length DOES NOT EXIST. You can make your enclosure longer and narrower, by keeping the same area.
The correct formulation is THIS :
What is the fencing length needed to construct a rectangle enclosure
containing 1800 ft^2 using a river as a natural boundary on one side?
2. The "solution" by @josgarithmetic is TOTALLY WRONG, starting from its third line to the end.
So you better simply IGNORE it.
Below find my correct solution.
xy = 1800 (1)
x + 2y -----> minimize (x is the length along the river)
So your task is to minimize (x+2y) under the given condition/restriction (1).
From (1), x = , so we need to minimize the function f(y) = .
The derivative f'(y) = - + 2.
To find the minimum of f(y), equate its derivative to zero
- + 2 = 0
= 2
= = 900
y = = 30.
ANSWER. The minimum fencing is at y = 30 ft perpendicular to the river and x = = = 60 ft along the river.
Solved.