CAIE P3 2017 June — Question 5

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2017
SessionJune
TopicDifferential equations

5 In a certain chemical process a substance \(A\) reacts with and reduces a substance \(B\). The masses of \(A\) and \(B\) at time \(t\) after the start of the process are \(x\) and \(y\) respectively. It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 0.2 x y\) and \(x = \frac { 10 } { ( 1 + t ) ^ { 2 } }\). At the beginning of the process \(y = 100\).
  1. Form a differential equation in \(y\) and \(t\), and solve this differential equation.
  2. Find the exact value approached by the mass of \(B\) as \(t\) becomes large. State what happens to the mass of \(A\) as \(t\) becomes large.