HELP! There is a line through the origin that divides the region bounded by the parabola y = 2x − 8x^2 and the x-axis into two regions with equal area. What is the slope of that line?

Answer 1

Start by sketching that parabola. The basic form is y = -x^{2 ,},which means it opens down. It may be easier to plot if you put it into a standard form---eg y + b = k(x + a)²

The area between the curve and the x-axis is found by integrating the difference between the given function and the function y = 0. You can use your plot to find the integration limits.

Next, write an equation for the difference between the given function and the line y = m*x. Set the integral = to 1/2 of the area found above,

Answer 2

As noted above this problem is NOT simple. I have solved but I used DESMOS to do it.

First, you need to find the total area involved which is the integral of 2x-8x² between 0 and 1/4; this is 1/48.

Second, you need to define the point where y=ax and y = 2x-8x² intersect. This happens when x= (2-a)/8. Let x₀ = (2-a)/8

I elected to calculate the area on the bottom (and right) of the figure. This consists of a triangle and a sliver (area between x₀ and 1/4).

I used DESMOS with a as a slider to compute this area...which needs to be 1/96.

The value for a evaluated in this way is between .41 and .42.

If you find a better way to do this when you review this problem, please let me know!