A chemical reaction produces a substance that spreads across a filter paper. The area of the paper, A cm2A\text{ cm}^2A cm2, covered by the substance ttt hours after the reaction starts is modeled by the differential equation
dAdt=A324t2,t>0 \frac{\text{d}A}{\text{dt}} = \frac{A^{\frac{3}{2}}}{4t^2}, \quad t > 0 dtdA=4t2A23,t>0Given that A=4A = 4A=4 when t=2t = 2t=2,
show that
A=(ptqt+r)2 A = \left( \frac{pt}{qt + r} \right)^2 A=(qt+rpt)2where ppp, qqq, and rrr are integers to be found.
According to the model, find the limiting value of the area covered as t→∞t \to \inftyt→∞.
