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October 25th, 2013, 12:04 PM   #11
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Re: Largest possible inscribed triangle in a circle

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Originally Posted by Daltohn
Since we're on the topic of optimization, how do you determine the largest value the expression (3x+4y) can take, when you have the constraint x^2+y^2=14x+6y+6 for the coordinates (x,y)?

The answer is 73, I can get that using substitution and then the derivative or Lagrange Multipliers. But how do you determine that without any calculus at all? Is there some trick since the constraint is a circle? It's a problem from a while back, when we hadn't done any calculus.
One method is to substitute w = x - 7 and z = y - 3, then (w, z) lies on a circle centre (0,0) with radius 8. And, switching to polar coordinates, w = 8cos(a) and z = 8sin(a).

So we have to maximise 24cos(a) + 32sin(a) + 33.



This has a maximum of 40, hence the maximum of the function is 73.
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October 27th, 2013, 08:19 AM   #12
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Re: Largest possible inscribed triangle in a circle

Thanks a lot for your help, everyone!
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