Joe M. answered • 05/23/19
College Professor with AP Calculus Experience
Hi Ariel,
I’ll assume that you are comfortable doing calculations with integrals, please message me if you need more description on this.
The graph of y = 2x - 7x2 is a parabola opening downward. It crosses the x-axis when y = 0:
0 = 2x - 7x2
0 = x(2 - 7x)
So the x-intercepts are when x = 0 and x = 2/7.
The total area of the bounded region is then the integral of 2x - 7x2 from x = 0 to x = 2/7. After doing the antiderivative to get x2 - (7/3)x3 we plug in the boundaries and get (2/7)2 - (7/3)(2/7)3 = 4/147. When we cut this region in half that area is 2/147.
The line through the origin will intersect the parabola at the origin. It will also intersect at some point where x = a; it must also be true at that point the y-value is 2a - 7a2. Since the line runs through the points (0,0) and (a, 2a - 7a2), then the slope of that line is (2a - 7a2 - 0)/(a - 0) = 2 - 7a. So the equation of the line is y = (2 - 7a)x.
Since the line splits the bounded region in half, we are ready to do integration to find the area of the top half. The two points of intersection are x = 0 and x = a. The half-region of interest is the one with the parabola on top and the line on the bottom. So our integration is 2x - 7x2 (the top) minus (2 - 7a)x (the bottom) from x = 0 to x = a:
Antiderivative of 2x - 7x2 - (2 - 7a)x =
Antiderivative of 2x - 7x2 - 2x + 7ax =
Antiderivative of -7x2 + 7ax =
(-7/3)x3 + (7a/2)x2
When we plug in the lower bound x = 0 we just get 0 so can ignore it. When we plug in a we should get the 2/147 we found earlier, so we can solve for a:
(-7/3)a3 + (7a/2)a2 = 2/147
(-7/3)a3 + (7/2)a3 = 2/147
(7/6)a3 = 2/147
a3 = 4/343
a = (cube root of 4)/7.
Since we know a, and had earlier found that the slope of the line is 2 - 7a, then the slope of the line is:
2 - 7[(cube root of 4)/7] =
2 - (cube root of 4) Final Answer
Please let me know if I can be more helpful, thanks!
Joe
