 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/6/a/solution

4 π R^{2} = 400 π m^{2}

. Dividing the given surface-patch area by that total gives the dimensionless fraction

\frac{10}{400 π} = \frac{1}{40 π} \approx 0.00796,

or about $0.796%$ of the surface. The patch area is already measured on the surface, so no planar projection or spherical-cap conversion is needed.

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