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Math Help - Optimizations question with cylinder in sphere with a radius of 5

  1. #1
    s3a
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    Optimizations question with cylinder in sphere with a radius of 5

    Question: A right cylinder is inscribed in a sphere with a radius of 5. Determine the maximum volume of the cylinder.

    Answer: Volume ≈ 302

    I'm having problems visualizing and even drawing this thing in the first place. The diameter of the sphere is not equal to the height cylinder, right? Apparently, I need to construct a right angle triangle in there but what do I do then?

    Any help would be GREATLY appreciated!
    Thanks in advance!
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    This one is kinda' related. Read through it and see if you can construct a route to your problem:

    http://www.mathhelpforum.com/math-he...al-volume.html
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    Quote Originally Posted by s3a View Post
    Question: A right cylinder is inscribed in a sphere with a radius of 5. Determine the maximum volume of the cylinder.

    Answer: Volume ≈ 302

    I'm having problems visualizing and even drawing this thing in the first place. The diameter of the sphere is not equal to the height cylinder, right? Apparently, I need to construct a right angle triangle in there but what do I do then?

    Any help would be GREATLY appreciated!
    Thanks in advance!
    If you use symmetry then you can visualize it as a rectangle inscribed in a circle, the revolve the about the z-axis.

    For a certain radius of cylinder r, the height will touch the sphere r^2 + z^2 = 25 \; \text{or}\; z = \sqrt{25-r^2} (X2 for the full height)

    Thus, what you wish to maximize is

     <br />
V = 2\pi r^2 \sqrt{25-r^2}<br />
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    Quote Originally Posted by s3a View Post
    Question: A right cylinder is inscribed in a sphere with a radius of 5. Determine the maximum volume of the cylinder.

    Answer: Volume ≈ 302

    I'm having problems visualizing and even drawing this thing in the first place. The diameter of the sphere is not equal to the height cylinder, right? Apparently, I need to construct a right angle triangle in there but what do I do then?

    Any help would be GREATLY appreciated!
    Thanks in advance!
    1. See attachment.

    2. With your question R = 5.

    3. The volume of a cylinder is V = \pi r^2 \cdot h

    r^2 + \frac14 h^2 = R^2~\implies~r^2 = R^2 - \frac14 h^2

    4. You'll get the volume of the cylinder as a function wrt h:

    V(h) = \pi R^2 \cdot h - \frac14 \pi h^3

    5. Differentiate V(h) wrt h and solve the equation V'(h) = 0 for h. I've got h = \frac23 \sqrt{3} \cdot R (Of course there is a negative solution too, but a negative radius of a sphere isn't very plausible)

    6. Plug in this value into the equation of the function:

    V\left(\frac23 \sqrt{3} \cdot R\right) = \pi R^2 \cdot \frac23 \sqrt{3} \cdot R - \frac14 \left( \frac23 \sqrt{3} \cdot R \right)^3

    7. The exact value is V = \frac{500}9 \pi \sqrt{3}
    Attached Thumbnails Attached Thumbnails Optimizations question with cylinder in sphere with a radius of 5-zyl_inkugl.png  
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    Quote Originally Posted by earboth View Post
    1. See attachment.

    2. With your question R = 5.

    3. The volume of a cylinder is V = \pi r^2 \cdot h

    r^2 + \frac14 h^2 = R^2~\implies~r^2 = R^2 - \frac14 h^2

    4. You'll get the volume of the cylinder as a function wrt h:

    V(h) = \pi R^2 \cdot h - \frac14 \pi h^3

    5. Differentiate V(h) wrt h and solve the equation V'(h) = 0 for h. I've got h = \frac23 \sqrt{3} \cdot R (Of course there is a negative solution too, but a negative radius of a sphere isn't very plausible)

    6. Plug in this value into the equation of the function:

    V\left(\frac23 \sqrt{3} \cdot R\right) = \pi R^2 \cdot \frac23 \sqrt{3} \cdot R - \frac14 \left( \frac23 \sqrt{3} \cdot R \right)^3

    7. The exact value is V = \frac{500}9 \pi \sqrt{3}
    This is much easier than mine same answer though.
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    Now, I get how to do it and I get the final answer! But...what is the restriction on r (cylinder's radius) to prove that it is indeed the maximum. Would the 1st and second derivative tests work as well? Or would those just tell me that it is a local maximum or minimum and not if it is the absolute maximum or minimum?
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    Quote Originally Posted by s3a View Post
    Now, I get how to do it and I get the final answer! But...what is the restriction on r (cylinder's radius) to prove that it is indeed the maximum. Would the 1st and second derivative tests work as well? Or would those just tell me that it is a local maximum or minimum and not if it is the absolute maximum or minimum?
    1. The 2nd derivative is negative for all positive values of h. Thus you have found a maximum.

    2. The radius of the cylinder varies between 0 and R: If r = 0 then the volume of the cylinder is 0, which is quite a minimal value for a volume. If r = R then the volume of the cylinder is 0 too, which is another minimal value for a volume. Since the V-function is continuous and differentiable you know that between 2 minimums (is that the correct expression? It sounds like a very small mother) is a maximum (big mother?)
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