Math Quiz 5

[Ether]

This one requires multiple skills and tools, including calculus.

Given:

1) y = b - a*x² (b>0, a>0, y>=0)

2) the total length of the curve is 10

Problem:

Find the value of b which maximizes the closed area between the curve and the x axis.

[Bryce Paputa]

Spoiler for arc length:

Well (I think) I've got (2 sqrt(4(ab)^2 + ab) + arcsinh(2 sqrt(ab))) / (2a) = 10 from the arc length restriction, but I have no idea how to get that in terms of either a or b. I'm pretty tired, so I'll take another look tomorrow sometime not after midnight. Once you have a nice relationship with a as a function of b or vice versa it should be easy to optimize. Maybe a nice function doesn't exist though, not sure.

Also, it's pretty easy to get a as a function of ab, I'll have to think about if that's actually helpful though. Might be able to get the inverse of it.

[Bryce Paputa]

Spoiler for solution?:

Ok, next I defined a variable c=ab, and got a as a function of c using the arc length. I also got area as a function of c. Then, because the derivative of the area is some insanely long expression, I put it into my CAS and got the answer ab = 1.515, a=.817 and b=1.853.

EDIT: somewhere I lost a factor of two, the actual answer is:

Spoiler for actual solution:

a=.626, b=4.06

[Ether]

Originally Posted by Bryce Paputa

a=.817 and b=1.853

Plug those numbers into your arc length expression and see if you get "10".

Originally Posted by Bryce Paputa

EDIT: somewhere I lost a factor of two, the actual answer is:
a=.626, b=4.06

Plug those numbers into your area expression and tell me what you get.

[Bryce Paputa]

Spoiler for area:

12.927. Also noticed that, approximately, a*sqrt(a) = 1/b * sqrt(b). Not sure if that's a coincidence or if figuring that out gives an easier way to solve the problem.

[Ether]

Originally Posted by Bryce Paputa

12.927

Double-check your area formula.

If you're sure it's correct, please post it.

[Bryce Paputa]

Nope, it was wrong. Correct area, a and b are

Spoiler for answer:

a=.285, b=3.34, area = 15.2

[Ether]

Originally Posted by Bryce Paputa

Nope, it was wrong. Correct area, a and b are
a=.285, b=3.34, area = 15.2

Nice job.

Want some reps? Please post your work and explain each step.

[Bryce Paputa]

Here's how I did it: https://docs.google.com/document/d/1...it?usp=sharing

[Ether]

Originally Posted by Bryce Paputa

Here's how I did it: https://docs.google.com/document/d/1...it?usp=sharing

Very interesting approach.

On page2 of the Google doc you say "Assume dR/dc=0, Solve for c"... But I don't see dR/dc anywhere, and I don't see you setting dR/dc=0 and solving analytically for c.

It looks like you gave a(c), b(c), and R(c) to desmos, and let it figure out dR/dc (numerically?) and find the zero crossing to get the desired value of c. Yes? I'm not familiar with the desmos calculator so I'm guessing.

If you would add just a bit more explanation to the Google doc for the benefit of future readers that would be most helpful.

[Bryce Paputa]

Originally Posted by Ether

Very interesting approach.

On page2 of the Google doc you say "Assume dR/dc=0, Solve for c"... But I don't see dR/dc anywhere, and I don't see you setting dR/dc=0 and solving analytically for c.

It looks like you gave a(c), b(c), and R(c) to desmos, and let it figure out dR/dc (numerically?) and find the zero crossing to get the desired value of c. Yes? I'm not familiar with the desmos calculator so I'm guessing.

If you would add just a bit more explanation to the Google doc for the benefit of future readers that would be most helpful.

That's correct, I did it numerically with the calculator. The Google doc now clarifies this. I'll look into an analytic solution.