There is a line through the origin that divides the region bounded by the parabola y = 8 x − 5 x 2 and the x-axis into two regions with equal area. What is the slope of that line?

Answer 1

Answer 2

Equation of line is y = mx;

y =8x - 5x² = 0; x = 0 or x = 1.6

mx = 8x - 5x²; x = 0 or x = (8 - m)/5;

∫_{(8 - m)/5}^1.6(8x - 5x²)dx = 1/2∫₀^1.6(8x - 5x²)dx;

(4x² - 5x³/3)_{(8 - m)/5}^1.6 = 0.5(4x²- 5x³/3)₀^1.6;

-[4(8- m)²/25 - (8 - m)³/75] + 256/75 = 128/75

12(8 - m)² - (8 - m)³ = 128

8 - m = 4; m = 4