An industrial spill of a viscous lubricant is spreading across a flat floor. The area of the spill, L m2L \text{ m}^2L m2, at time ttt hours after the spill is discovered, is modelled by the differential equation
dLdt=LL5t2,t>0 \frac{dL}{dt} = \frac{L\sqrt{L}}{5t^2}, \quad t > 0 dtdL=5t2LL,t>0Given that the spill covers 16 m216 \text{ m}^216 m2 at the moment it is discovered (t=1t = 1t=1),
show that
L=(ptqt+r)2 L = \left( \frac{pt}{qt + r} \right)^2 L=(qt+rpt)2where ppp, qqq, and rrr are integers to be found.
According to the model, find the limiting value of the area covered by the spill as t→∞t \to \inftyt→∞.
