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

Question

what is the slope of the line?

Answer 1

Hi Ariel,

Let y=kx be the equation of the line...we are going to find k

Find the intersection of the line with the parabola kx=4x-2x² the x value where this is true is λ, λ=(4-k)/2.

Next we find the area under the parabola and above the line call this area A1=∫((4x-2x²)-kx)dx from 0 to λ,

A1= λ³/3

Next we find the area under the parabola and the x axis call this area AT=∫(4x-x²)dx from 0 to 2=8/3

We know that A1+A2=AT so A2=AT-A1=8/3-λ³/3= (8-λ³)/3

We want A1=A2 so we must have λ³=8-λ³ solving for λ gives λ=(4)^1/3=1.59 we can now find k because λ=(4-k)/2 and we know λ so k=.825 which is the slope of the line that divides the area in two.

Hope this helps let me know if you have quesstions

Regards

Jim