a straight section of railroad track crosses two highways at points that are 400m and 600m, respectively, from an intersection. determine the dimensions of the largest rectangular lot that can be laid out in the triangle formed by the railroad and highways.

Answer 1

Laid out on the x-y plane, the track's line can be described by

y = 400 - 2/3 x

So, if the rectangle has one corner on the line and the opposite corner at (0,0), its area is

a = xy = x(400 - 2/3 x) = 400x - 800/3 x^2

This is just a parabola. Find its vertex, and that is the maximum area.

I suppose you could also investigate rectangles whose sides are not parallel to the axes, but that can get quite complicated...