A rectangular lot is bounded on one side by a river and on the other three sides by 80 m of fencing. What is the dimensions of the largest possible lot?

1 Answer
Feb 25, 2017

#20"m" xx 40"m"#

Explanation:

If there were no river, but twice as much fencing, then the optimal rectangle would be a square #40"m" xx 40"m"#, having a perimeter of length #4xx40"m" = 160"m"#.

Given such a square, run a river through the middle of it, dividing the plot and the fencing in two to find the optimal rectangular plot dimensions: #20"m" xx 40"m"#

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Alternatively, we can solve this using a little algebra:

Let the two opposite sides have length #t# and the side opposite the river length #80-2t#.

Then the area is given by:

#a(t) = t(80-2t) = 80t-2t^2 = 2(400-400+40t-t^2) = 2(400-(20-t)^2)#

This takes maximum value when #(20-t)^2# has its minimum value: #0#, that is when #t=20#.

So the two equal sides are of length #20"m"# and the other side of length #80-2(20) = 40"m"#