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In a chemical reaction, compound B is formed from compound A and other compounds. The mass of B at time t minutes is x 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.

(i) Show that the model can be written as dx/dt = kx(1-x), where k is a constant. [1]

Initially, the mass of B is 0.2 kg.
(ii) Solve the differential equation, expressing x in terms of k and t. [7]

After 15 minutes, the mass of B is measured to be 0.9 kg.
(iii) Find the value of k, correct to 3 significant figures. [2]

(iv) Find the mass of B after 30 minutes. [1]

(v) Explain what the model predicts for the mass of A remaining for large values of t. [1]

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