Here is a video I made which illustrates an approach:
https://drive.google.com/file/d/1JGLoOfwJvOSZr_6KNIKoGla_yyf3l3yc/view
I have not narrated it. I can add narration, but first I recommend that you try to follow it and work out as much as you can. You can post work here and we can try to give you further guidance. The equation that you end of needing to solve is a rather ugly cubic equation. If part of the purpose of the problem is to practice solving cubics, then go for it. Otherwise there are calculators that can solve it.
Patrick F.
tutor
Let me clarify what are m and p. m is the gradient of the straight line. Our goal is to find m.
p is the x value when the line and the curve are equal. p can be found by setting mx = -7x^2+5x. p in terms of m is (5-m)/7
We set up the equation with the integrals in the final step. Basically, total area under the curve from 0 to 5/7 is equal to twice the area of the difference between the curve and the line for 0 to p. This equation is solved for m.
First, try to understand why we need p. Finding p in terms of m is not hard. Then understand the equations with integrals, and importantly the limits of the integrals.
What looks hard is possibly the solving of the cubic for m at the very end. I didn't have the patience so I used technology and got m=1.03. Looks reasonable when I draw the the line y = 1.03x
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01/18/23
Adam S.
After solving for m I get m is equal to either 5 or -2, and when solving for p, p is = to 0 or 3/2. I am not quite sure how I move forward from this step as I tried a couple ways of substituting but was not able to get a valid answer.01/18/23