The surface area of a sphere of radius $5\space cm$ is five times the area of the curved surface of a cone of radius $4\space cm$. Find the height of the cone.

Using the surface area of a sphere formula, $4\pi r^2$, the sphere's area is $4\pi(5^2) = 100\pi$. The curved surface area of a cone is $\pi r l$, which equals $100\pi / 5 = 20\pi$. Substituting the cone radius 4 gives $\pi(4)l = 20\pi$, so the slant height $l = 5$. Using the Pythagorean theorem for the cone, $h = \sqrt{l^2 - r^2} = \sqrt{5^2 - 4^2} = \sqrt{9} = 3$. The height of the cone is 3 cm.

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