9709 P31 - Jun 2018 - Q6
2289
In a certain chemical reaction, the amount, \(x\) grams, of a substance is decreasing. The differential equation relating \(x\) and \(t\), the time in seconds since the reaction started, is
\[ \frac{dx}{dt} = -\frac{kx}{\sqrt{t}}, \]
where \(k\) is a positive constant. It is given that \(x = 100\) at the start of the reaction.
- Solve the differential equation, obtaining a relation between \(x\), \(t\), and \(k\).
- Given that \(t = 25\) when \(x = 80\), find the value of \(t\) when \(x = 40\).
