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#1
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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. |
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#2
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Re: Math Quiz 5
Spoiler for arc length:
Last edited by Bryce Paputa : 29-10-2014 at 00:25. |
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#3
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Re: Math Quiz 5
Spoiler for solution?:
EDIT: somewhere I lost a factor of two, the actual answer is: Spoiler for actual solution:
Last edited by Bryce Paputa : 29-10-2014 at 18:33. |
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#4
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Re: Math Quiz 5
Plug those numbers into your arc length expression and see if you get "10".
Quote:
Last edited by Ether : 29-10-2014 at 18:46. |
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#5
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Re: Math Quiz 5
Spoiler for area:
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#6
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Re: Math Quiz 5
Double-check your area formula.
If you're sure it's correct, please post it. |
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#7
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Re: Math Quiz 5
Nope, it was wrong. Correct area, a and b are
Spoiler for answer:
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#8
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Re: Math Quiz 5
Quote:
Want some reps? Please post your work and explain each step. |
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#9
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Re: Math Quiz 5
Here's how I did it: https://docs.google.com/document/d/1...it?usp=sharing
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#10
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Re: Math Quiz 5
Quote:
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. |
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#11
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Re: Math Quiz 5
Quote:
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#12
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Re: Math Quiz 5
It may not exist. My CAS choked on it.
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#13
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Re: Math Quiz 5
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. |
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#14
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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.
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#15
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Re: Math Quiz 5
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. Last edited by Ether : 29-10-2014 at 23:33. |
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