Two vertices of a rectangle are on the positive x axis. The other two vertices lie on the lines y= 4x and y= -5x + 6. Then the maximum area of the rectangle is?

Answer 1

Draw a diagram. If the base of the rectangle is the segment (a,0) to (b,0) then the upper vertices are (a,4a) and (b,6-5b).

Since it is a rectangle, the height is the same, so 4a = 6-5b.
That is, b = (6-4a)/5

Now, the area is

(b-a)(4a) = ((6-4a)/5-a)(4a) = 12a/5 (2-3a)

This is a parabola with vertex at (1/3, 4/5).

check:
So, b=14/15

and the maximum area is

(14/15 - 1/3)(4/3) = (9/15)(4/3) = 4/5