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). Treating Earth as a sphere with center $O$ and radius $r = 6400$ km, the latitude of a point $A$ on its 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 $\mathrm { km } ^ { 2 }$). Then $S$ accounts for approximately what percentage of Earth's surface area? A. $26 \%$ B. $34 \%$ C. $42 \%$ D. $50 \%$
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). Treating Earth as a sphere with center $O$ and radius $r = 6400$ km, the latitude of a point $A$ on its 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 $\mathrm { km } ^ { 2 }$). Then $S$ accounts for approximately what percentage of Earth's surface area?
A. $26 \%$
B. $34 \%$
C. $42 \%$
D. $50 \%$