A crystalline deposit of mass MMM milligrams is formed on an electrode during a chemical process. At time ttt seconds after the process begins, the rate of increase of the mass is modeled by the differential equation
dMdt=M326t2,t>0 \frac{\text{d}M}{\text{dt}} = \frac{M^{\frac{3}{2}}}{6t^2}, \quad t > 0 dtdM=6t2M23,t>0Given that M=16M = 16M=16 when t=4t = 4t=4,
show that
M=(ptqt+r)2 M = \left( \frac{pt}{qt + r} \right)^2 M=(qt+rpt)2where ppp, qqq, and rrr are integers to be found.
According to the model, find the limiting value of the mass of the deposit as t→∞t \to \inftyt→∞.
