\section*{Problem 1 - 051221}

A spherical cap is cut from a sphere of radius \(r\), consisting of a cone of height \(h\) and the associated spherical segment.

a) What is the length \(h\) of the height of the cone if the area of the cut spherical cap is equal to one third of the surface area of the sphere?

b) What is the length \(h\) of the height of the cone if the volume of the spherical sector is equal to one third of the volume of the sphere?
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With the standard notations \(r\) for radius, \(h\) for height, and \(A\) for area or \(V\) for volume, we have:

a) \(\frac{1}{3} A_{\text {sphere }}=A_{\text {cap }} \rightarrow \frac{4}{3} \pi \cdot r^{2}=2 \pi \cdot r \cdot h_{\text {cap }}=2 \pi \cdot r \cdot(r-h)\)

From this, since \(r \neq 0: r-h=\frac{2}{3} r\) thus \(h=\frac{1}{3} r\)

b) \(\frac{1}{3} V_{\text {sphere }}=V_{\text {sector }} \rightarrow \frac{2}{3} \pi \cdot r^{2} \cdot(r-h)=\frac{4}{9} \pi \cdot r^{3}\) \(h=\frac{1}{3} r\)