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    Topics || Problems

    A cylinder is inscribed in a given sphere of radius a. Find the dimension of the cylinder if its lateral surface area is maximum.

    Solution:

    cylinder inscribed in a sphere

    The lateral surface area of the cylinder is \(A=2\pi r_c h\):

    Since the radius of the sphere is a constant, "\(a\)", rewrite the lateral surface area of the cylinder as a function of the its height or its radius with the radius of the sphere (\(a\)).

    \((\frac{h}{2})^2 +(r_c)^2=a^2\)

    \(r_c = \frac{1}{2} \sqrt{4a^2-h^2}\)

    \(A=2\pi r_c h\)

    \(A=2\pi (\frac{1}{2} \sqrt{4a^2-h^2}) h\)

    \(A=\pi (\sqrt{4a^2-h^2}) \sqrt{h^2}\)

    \(A=\pi \sqrt{4h^2a^2-h^4}\): The lateral surface area of the cylinder as a function of \(h\) and \(a\)

    Maximize the lateral surface area \(A\).

    \(\frac{dA}{dh}=\pi \frac{1}{2}\frac{8ha^2-4h^3}{\sqrt{4h^2a^2-h^4}}\)

    \(0=\pi \frac{1}{2}\frac{8ha^2-4h^3}{\sqrt{4h^2a^2-h^4}}\)

    \(0=8ha^2-4h^3\)

    \(h^2=2a^2\)

    \(h=a\sqrt{2}\)

    \(r_c=a\frac{\sqrt{2}}{2}\)

    Thus the dimension of the cylinder (at maximum lateral surface area) inside a sphere of radius \(a\) is:

    \(h=a\sqrt{2}\)

    \(r_c=a\frac{\sqrt{2}}{2}\)