9709 P3 - Jun 2003 - Q7
2314
In a chemical reaction a compound X is formed from a compound Y. The masses in grams of X and Y present at time t seconds after the start of the reaction are x and y respectively. The sum of the two masses is equal to 100 grams throughout the reaction. At any time, the rate of formation of X is proportional to the mass of Y at that time. When t = 0, x = 5 and \(\frac{dx}{dt} = 1.9\).
(i) Show that x satisfies the differential equation \(\frac{dx}{dt} = 0.02(100 - x)\). [2]
(ii) Solve this differential equation, obtaining an expression for x in terms of t. [6]
(iii) State what happens to the value of x as t becomes very large. [1]
