Rate of Change of Slice Area
A sphere has a radius of 10 units, and is centered at the origin. A sliding plane is defined by
\[ -\sin(\pi / 3) (y + 12 - (24/50) t) + \cos( \pi / 3 ) z = 0 \]
For a certain range of \( t \) this plane crosses the sphere, and the intersection is a circle whose area is a function of \( t \). Find the rate of change of the cross-sectional area at \( t = 15 \).