9. In a chemical reaction, compound B is formed from compound A and other compounds. The mass of B at time \(t\) minutes is \(x \mathrm {~kg}\). The total mass of A and B is always 1 kg . Sadiq formulates a simple model for the reaction in which the rate at which the mass of \(B\) increases is proportional to the product of the masses of \(A\) and \(B\).

1. Show that the model can be written as \(\frac { \mathrm { d } x } { \mathrm {~d} t } = k x ( 1 - x )\), where \(k\) is a constant. Initially, the mass of B is 0.2 kg .
2. Solve the differential equation, expressing \(x\) in terms of \(k\) and \(t\). After 15 minutes, the mass of B is measured to be 0.9 kg .
3. Find the value of \(k\), correct to 3 significant figures.
4. Find the mass of B after 30 minutes.
5. Explain what the model predicts for the mass of A remaining for large values of \(t\).
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