I still don't get how to work out the similar triangles one

Two more questions coming through, (attached)

Q1. In the solutions, after obtaining 2 different angles, we are to differentiate and then choose the one where x>0. I don't understand how to decide to differentiate to check for the right solution. How do you get to that step? What thinking is involved?

Q2. Why can't we integrate (3)^2 - (e^2y)^2 ?

Thanks so much for answering my questions especially when you have trials study going on!!

For the first question I assume you got this?

I don't think you really need to differentiate here. Each value of m, when substituted back into the equation from part i) should yield a different parabola both of which are incidentally perfect squares. One value of m has one positive root and the other has one negative root, and it's pretty obvious that the root should be positive since the plane touches the hill at 2<x<6. Then you choose the correct value of m, then use m=tan theta.

Not sure what you mean by your second question (my brain is fizzing out right now, I might get back to this later) but this is what I get for the volume:

For the similar triangles one, AB is parallel to CE and AD is parallel to BC. In triangles EFD and EBC you have two corresponding angles in parallel lines, so they're equiangular, and thus similar. In triangles ABF and EDF, you have two alternate angles in parallel lines, so they're equiangular and thus similar. Since CD:DE is 2:1, BF:FE must also be 2:1 since similar triangles have corresponding sides in the same proportion as other corresponding sides. Since BF:FE = 2:1, AF:FD = 2:1 as well for the same reason. Then AF is 2/3 of AD, which is equal to BC. ie. AF:BC = 2:3

Hope this helps