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Math Help - Calculus word problem, maximization

  1. #1
    Senior Member maxpancho's Avatar
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    Calculus word problem, maximization

    If you fit the cone with the largest possible surface area (lateral area plus area of base) into a sphere, what percent of the volume of the sphere is occupied by the cone
    Couldn't really put the constraints on the formula for surface area of a cone.


    So:

    $V(sphere) = \dfrac{4}{3}\pi r^3$
    I guess I could just take $r = 1$ then it is: $V(sphere) = \dfrac{4}{3} \pi$

    $ A(cone)= \pi r (r + \sqrt{h^2 + r^2}) $

    And here we have to define the domain, as I understand; so $ r \leq 1$, $h \leq 2r$


    Is this correct thinking?



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  2. #2
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    Re: Calculus word problem, maximization

    What reason do you have to think that the height of the cone must be less than or equal to the radius of its base? Of course, h cannot be larger than the diameter of the sphere but that is '2', not '2r'.
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    Senior Member maxpancho's Avatar
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    Re: Calculus word problem, maximization

    Oh, well, I guess it should be 2 if we already had r=1.

    But what do I do next? I can't seem to solve it. As usual take the derivative and set it equal to zero?
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    Re: Calculus word problem, maximization

    Quote Originally Posted by maxpancho View Post
    Oh, well, I guess it should be 2 if we already had r=1.

    But what do I do next? I can't seem to solve it. As usual take the derivative and set it equal to zero?
    I think you are confusing yourself! Are you using "r" to mean the radius of the sphere or the radius of the base of the cone? That was my point.
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  5. #5
    Senior Member maxpancho's Avatar
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    Re: Calculus word problem, maximization

    Yes, you're right.

    Well, let's say then:

    $V(sphere) = \dfrac{4}{3}\pi R^3$
    $R = 1$
    $V(sphere) = \dfrac{4}{3} \pi$



    $ A(cone)= \pi r (r + \sqrt{h^2 + r^2}) $

    Domain: $ r \leq 1$, $h \leq 2r$ or $ h \leq 2 $


    HallsofIvy, could you help me go from here? I'm not sure what to do next. I have two variables in the function for surface area of a cone, and I can't seem to extract any more info from the description of the problem.
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    Re: Calculus word problem, maximization

    The first thing I would do is draw a picture! Draw a circle (representing a side view of the sphere) and draw an isosceles triangle in the circle (representing the cone inside the sphere). Use that to find a relation involving h, the height of the cone, r, the radius of the base of the cone, and R, the radius of the sphere.
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  7. #7
    Senior Member maxpancho's Avatar
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    Re: Calculus word problem, maximization

    Oh, okay, I had done it before, but only now realized that I could use the equation of a circle as a constraining factor. And... I got the correct answer.

    Thanks. Sometimes it's good to be able to rely on the authority of an experienced person\mathematician while you're still learning the subject.
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