Thread: areas between curves

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

I can't figure it out. I don't know how to start

Originally Posted by zaddie21

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

I can't figure it out. I don't know how to start

calculate the area in quad I bounded by the parabola and the x-axis, call that area $\displaystyle A$.

let the line be $\displaystyle y = mx$

find the intersection of the line and parabola in quad I ...

$\displaystyle mx = 8x - 2x^2$

$\displaystyle 0 = (8-m)x - 2x^2$

$\displaystyle 0 = x(8 - m - 2x)$

x-values of intersection are $\displaystyle x = 0$ and $\displaystyle x = \frac{8-m}{2}$


$\displaystyle \frac{A}{2} = \int_0^{\frac{8-m}{2}} (8x-2x^2) - mx \, dx$

evaluate the integral and solve for $\displaystyle m$ ... have fun with the algebra.