There is a line through the origin that divides the region bounded by the parabola y=6x-5x^2 and the x-axis into two regions with equal area. What is the slope of that line?

Answer 1

First, let us where the parabola intesects the x-axis. To do this, we need to solve for y = 6x - 5x² = 0, which has 2 roots, x = 0 and x = 1.2. So, the area bounded by the parabola and the x-axis is :

A = ∫₀^1.2 dx (6x - 5x²)

Now, the question states that there is a line through the origin that divides this area into two equal parts.

A line through the origin has the form y = mx. First, let us find where this line intersects the parabola.

To do this, we need to solve the equation mx = 6x - 5x² ⇒ x (5x + m - 6) = 0 which has two roots : x = 0 and x = (6 - m)/5. So, the points of intersection of the line and the parabola are (0,0) and (x₀ , mx₀), where we have defined x₀ = (6-m)/5.

Let's draw a diagram to keep track of things (diagram can be accessed at https://postimg.cc/fVR8rfgd , because WyZant interface is too basic to upload diagrams without hitting the character limit :():

The line y = mx divides the area A into two parts; the part above (shaded purple) and the rest (blue + green). We are told that the area of each of these two parts = A/2.

How do we find the area of the purple shaded part?

First, let us just find the area bounded by the parabola and the x-axis between x = 0 and x₀; This is just the integral of the equation of the parabola from 0 to x₀.

The diagram shows that this includes both the purple and the blue shaded parts. So, in order to find the area of the purple shaded part, we must subtract the area of the blue shaded part from the integral we found above.

The blue shaded part is just a right angle triangle of base x₀ and height mx₀ .

So, putting everything together, the equation to be solved is :

area of the purple part = ∫₀^x0 dx (6x - 5x²) - mx₀²/2 = A/2, where x₀ = (6-m)/5

which is a cubic equation in m

Hope this helps.

Answer 2

1) Integrate the function from 0 to 1.2. This is the area below the parabola and the x-axis.

2) Divide that number by 2.

3) That number = ∫₀^x (6x - 5x²) dx

4) Use the value of x to determine the slope.