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Math Help - Maximisation Problem - Cone/Sphere

  1. #1
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    Maximisation Problem - Cone/Sphere

    Problem:
    What is the volume of the largest sphere that can be placed inside a cone with a slant height of 10?


    What I've done so far:
    • Recognize that finding the largest radius will suufice; once I find it, I can find the volume easily.
    • Reduce the problem to a right triangular cross-section with an inscribed semicircle. the semicircle has a diameter that lies on the height of the triangle. A radius has been drawn to the right angle vertex of the triangle, and to the hypotenuse of ten (at a right angle).

    If I call the far angle across from the semicircle \theta, then:
    10 sin(\theta) = r + \frac{r}{cos(\theta)}
    Or, 5 sin(2\theta) = rsin(\theta) + r
    Past this, I am stumped. An Implicit derivative doesn't seem to get me anywhere, but it is impossible to fully separate my two variables. Any thoughts?
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  2. #2
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    Re: Maximisation Problem - Cone/Sphere

    let \theta be the base angle of the cone.

    Note that a line from the center of the circle to the vertex of \theta bisects \theta.

    R = r \tan\left(\frac{\theta}{2}\right)

    r = 10\cos{\theta}


    R = 10\cos{\theta} \cdot \tan\left(\frac{\theta}{2}\right)

    find \frac{dR}{d\theta} and maximize ...

    I get the value \theta \approx 52^\circ

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