Can anyone help me with this?

Question

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

Answer 1

First draw it out and get the area bounded by the entire region (the area between the curve and the x axis)

We'll call this area A

Next, we have a line through the origin that splits that region in half. This line can be written as y = mx where we need to figure out what m is.

The area between the curve and the line must be (1/2)A.

Thus ∫_a^b 8x - 3x² - mx dx = (1/2) A

Now we need the interval. Looking at the graph of the curve and the line, we see that a = 0

b is going to be the other place where the line and curve intersect or more specifically the x coordinate of the place of intersection.

We'll find that intersection by setting the function for the curve and the line equal to each other and solving for x.

8x - 3x² = mx

3x² - 8x - mx = 0

3x² - (m+8)x = 0

x (3x - (m+8)) = 0

3x - (m+ 8) = 0

3x = m + 8

x = (m+8) / 3

Therefore b = (m+8)/ 3

Notice that b is in terms of m. This makes sense since the slope of the line is going to affect where the line intersects the curve

Thus we have ∫₀^(m+8)/3 8x - 3x² - mx dx = (1/2) A

Now, just solve for m.