I need understanding a calculus question:
Johnny Bravo is building a head board with the outline of the $y = x – x^2$, which is bounded by the $x$-axis. He wants to split the head board into half green and half red, however, he does not want to split it straight down the middle. Johnny wants to draw a diagonal line starting at the origin that will divide the headboard in half. What is the equation of the line?
My approach was to try and find the intersection point (x-value) of the function that will set the two areas equal to each other.
I set up the integral equations:
$\int_{-1}^{x}x – x^2 dx = \int_{x}^{0}x-x^2 dx$
However, I do not think that this is the correct approach – it did not get me anywhere.
The correct answer is:
$y = (1 – \frac{1}{2^{1/3}})x$
Does anyone know how to arrive at this answer?
Best Answer
the area of the head board
$\int_0^1 x - x^2 \ dx = \frac 16$
line from the origin $y = kx$
intersects the curve $y = x - x^2$
$kx = x-x^2\\ (1-k) x - x^2 = 0$
$\int_0^{1-k} (1-k)x - x^2 \ dx = \frac 1{12}$
$\frac {(1-k)^3}{6} = \frac 1 {12}$
$(1-k)^3 = \frac 12\\ k = 1-(\frac 12)^{\frac 13}\\ y = (1-(\frac 12)^{\frac 13})x$