4. The BeiDou-3 Global Navigation Satellite System is an important achievement of China's space program. In satellite navigation systems, geostationary satellites orbit in the plane of Earth's equator at an orbital altitude of 36,000 km (orbital altitude is the distance from the satellite to Earth's surface). Consider Earth as a sphere with center $O$ and radius $r = 6400$ km. The latitude of a point $A$ on Earth's surface is defined as the angle that $OA$ makes with the equatorial plane. The maximum latitude at which a geostationary satellite can be directly observed from Earth's surface is $a$. The surface area covered by the satellite signal on Earth's surface is $S = 2 \pi r ^ { 2 } ( 1 - \cos a )$ (in $\mathrm { km } ^ { 2 }$). What percentage of Earth's total surface area does $S$ represent? ( )
A. $26 \%$
B. $34 \%$
C. $42 \%$
D. $50 \%$
【Answer】C 【Solution】 【Analysis】From the given information, use the provided surface area formula and the formula for the surface area of a sphere to calculate the result. 【Detailed Solution】From the given information, the percentage of $S$ relative to Earth's surface area is approximately: $\frac { 2 \pi r ^ { 2 } ( 1 - \cos a ) } { 4 \pi r ^ { 2 } } = \frac { 1 - \cos a } { 2 } = \frac { 1 - \frac { 6400 } { 6400 + 36000 } } { 2 } \approx 0.42 = 42 \%$ . Therefore, the answer is: C.