Optimisation: volume of a cone

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YES.
You're told you have 'a right triangle with hypotenuse of length $a$'.
That means the length is constant. The value is not specified now and it's hidden behind the symbol $a$, but it is some constant.

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MisterMysterious
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MisterMysterious

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Updated on April 13, 2020

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  • MisterMysterious
    MisterMysterious over 2 years

    Question: A right triangle with hypotenuse of length a is rotated about one of its legs to generate a right circular cone. Find the greatest possible volume of such a cone.

    My thoughts:

    Given the formula for the volume of a cone and the Pythagorean Theorem, I can eliminate either the base or the height variable from the formula for the volume of a cone, giving me volume in terms of hypotenuse and height or volume in terms of hypotenuse and base.

    I think that the only way to determine the greatest possible volume of such a cone is to assume that the length of the hypotenuse (a) is constant, as I will otherwise be unable to find the derivative (dV/dh or dV/db) and then determine the maximum.

    Is this the case? Should I treat the length of hypotenuse as a constant?

    • amd
      amd over 4 years
      There’s no need for you to assume that the hypotenuse length is constant. That’s a given in the problem.
    • user
      user over 4 years
      @MisterMysterious Please remember that you can choose an answer among the given if the OP is solved, more details here meta.stackexchange.com/questions/5234/…
  • MisterMysterious
    MisterMysterious over 4 years
    Thanks a lot. Wasn't really confident with the wording. That clears it up.