Find the area of the largest rectangle having one side on the x-axis and inscribed in a triangle formed by the lines y=x, y=0 and 3x+y=20.

Answer 1

If the rectangle has height x, then it has vertices

(x,0), (x,x), ((20-x)/3,x), ((20-x)/3,0)

so, its area is
a = x((20-x)/3-x) = 20x/3 - 4x^2/3
da/dx = 20/3 - 8x/3
da/dx=0 at x=5/2, and

a(5/2) = 25/3

Answer 2

Thank you so much!