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
Explanation:
If there were no river, but twice as much fencing, then the optimal rectangle would be a square
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:
Alternatively, we can solve this using a little algebra:
Let the two opposite sides have length
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
So the two equal sides are of length