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Ether 28-10-2014 13:04

Math Quiz 5
 

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 29-10-2014 00:17

Re: Math Quiz 5
 
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 29-10-2014 17:49

Re: Math Quiz 5
 
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 29-10-2014 18:39

Re: Math Quiz 5
 
Quote:

Originally Posted by Bryce Paputa (Post 1406337)
a=.817 and b=1.853

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


Quote:

Originally Posted by Bryce Paputa (Post 1406337)
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 29-10-2014 18:57

Re: Math Quiz 5
 
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 29-10-2014 19:07

Re: Math Quiz 5
 
Quote:

Originally Posted by Bryce Paputa (Post 1406349)
12.927

Double-check your area formula.

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




Bryce Paputa 29-10-2014 19:25

Re: Math Quiz 5
 
Nope, it was wrong. Correct area, a and b are
Spoiler for answer:
a=.285, b=3.34, area = 15.2

Ether 29-10-2014 19:50

Re: Math Quiz 5
 
Quote:

Originally Posted by Bryce Paputa (Post 1406354)
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 29-10-2014 20:14

Re: Math Quiz 5
 
Here's how I did it: https://docs.google.com/document/d/1...it?usp=sharing

Ether 29-10-2014 22:20

Re: Math Quiz 5
 
Quote:

Originally Posted by Bryce Paputa (Post 1406359)

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 29-10-2014 22:28

Re: Math Quiz 5
 
Quote:

Originally Posted by Ether (Post 1406372)
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.

Ether 29-10-2014 22:39

Re: Math Quiz 5
 
Quote:

Originally Posted by Bryce Paputa (Post 1406374)
I'll look into an analytic solution.

It may not exist. My CAS choked on it.



Ether 29-10-2014 23:01

Re: Math Quiz 5
 
2 Attachment(s)
Quote:

Originally Posted by Ether (Post 1406376)
It may not exist. My CAS choked on it.

Using your R(c) method, a numerical root finder quickly converges to a precise value for c.

But you can get the same result without differentiating R(c), by simply plotting R(c) and locating the extremum.




Bryce Paputa 29-10-2014 23:09

Re: Math Quiz 5
 
I get 3 arcsinh(2 sqrt(c)) * sqrt(4c^2+c) = 2c+8c^2. I doubt there is a nice closed form of this.

Ether 29-10-2014 23:22

Re: Math Quiz 5
 
1 Attachment(s)

For completeness, here's how to solve the problem using constrained nonlinear optimization.

It's a bit easier to set up. All you need is the length and area as a function of a and b.

The area is the objective (to be maximized), and the length is a constraint on the values of a and b.




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