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Problem:
Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder whose volume is a maximum.
This is a calculus maxima problem. Having drawn a diagram I noticed that
The generic cylinder volume formula is
After that I did
. Set . Solve for h to get the max volume which does not match the answer of .Was I wrong in the initial observation R = b + h ??
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Let me revise my last post.
Working with my previous equation
yieldedSo I decided to solve for the cylinder radius instead with
Shouldn't the answer workout correctly regardless of whether I solve for the radius or the height? Why did it work out solving for the radius? How can I know in advance whether to solve for the radius or the height?
Last edited by leadfoot (2010-11-07 17:36:04)
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You can get the correct answer by solving for the height, you just aren't being careful with what you're solving for. When you solved for
you got 2/3 R, which is the radius. However, if you solved you probably got 1/3 R, right? But remember that this answer is in terms of the height of the cylinder. Remembering that b + h = R, you get b = R - 1/3 R = 2/3 R, which is what you got when you solved for the radius directly.Wrap it in bacon
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... if you solved
you probably got 1/3 R, right?
I didn't get 1/3 R.
Did I make a math error to arrive at a quadratic? Is factoring possible without knowing the true value of R?
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You know how to use the quadratic formula right? h is your variable, so a = 3, b = -4R, and c = R^2. If you solve for this you should get h = 1/3 R. Although this particular equation is easy to factor too: (3h - R)(h - R) = 0.
Last edited by TheDude (2010-11-08 05:37:10)
Wrap it in bacon
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of course you're right. i forgot about the quadratic formula.
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