A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river (see the figure). What are the dimensions of the field of largest area that he can fence?

Question

(a) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the dimensions of the largest possible field. (Enter your answers as a comma-separated list.)

Reiny · Answer

Length of side parallel to river, (we only have one such side) ---- y ft
width of rectangle --- x ft
so 2x + y = 2400
or y = 2400 - 2x

area = xy = x(2400 - 2x)
= -2x^2 + 2400x

this is represented by a parabola which opens downwards,
the x of the vertex is -2400/-4 = 600
then y = 2400 - 2(600) = 1200 ft

oobleck · Answer

Note that as in all of these problems, no matter how the area is divided up into sections, the maximum area is achieved when the available fencing is divided equally between lengths and widths. Here, there are 2400 ft of fencing, so 1200 (1x1200) will be lengths, and 1200 (2x600) will be widths.