There is a line through the origin that divides the region bounded by the parabola y=3x-4x^2 and the x-axis into two regions with equal area. What is the slope of that line?

Answer 1

The unknown which we are trying to define here is the slope of the line which I will define as m.

The roots of y=3x-4x² are 0 and 3/4 so we know that both curves cross the origin (0,0).

If the line y=mx divides the region bounded by y=3x-4x² and the x-axis such that they have equal area then by definition the area bounded by y=3x-4x² and y=mx is exactly half of the region bounded by y=3x-4x² and the x-axis.

Lets first find the area bounded by y=3x-4x² and the x-axis.

A = ∫3x-4x²dx from 0 to 3/4

A = (3/2)x² - (4/3)x³ evaluated between 0 and 3/4

A = 27/32 - 108/192

A = 9/32

Now that we have the whole area then the area bounded between the curves y = 3x - 4x² and y=mx must be half of this. I'll call this new area A₂.

A₂ = 9/64

Now it is necessary to know where the curves y=3x-4x² and y=mx intersect. We've already determined that they intersect at (0,0). Now we just need the x-coordinate for the other intersection point. We just have to set the equations equal to themselves and solve for x.

3x -4x² = mx

Factor out an x and solve

x = (3-m)/4

Now we are finally ready to solve for m. Because we know the area of A₂ we can set up the following equation and solve for m.

∫(3-m)x - 4x²dx from 0 to (3-m)/4 =9/64

((3-m)/2)x² -(4/3)x³ evaluated between 0 and (3-m)/4 = 9/64

(3-m)³/32 -4(3-m)³/192 = 9/64

2(3-m)³ =27

m = 3 - (27/2)^1/3 ≈ 0.6189

If you'd like to check this value yourself you can set up 2 new integrals representing the 2 split areas. Both areas should now be equal. Hope this helps.

-Eric

Answer 2

In my opinion there is no easy (non-messy) way to solve this problem.

You must first determine the area of the region enclosed by the parabola and the x-axis and divide it in half; call the value A.

The intersection of the straight line with the parabola is where m=3-4x₀.

The bottom half area is then given by .5x₀(3-4x₀)x₀ + the integral from x₀ to 3/4 of (3x-4x²).

Then set these 2 areas equal and solve for x₀. Good luck!