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Problem: In a chemical reaction, a compound X is formed two compounds Y and Z. The masses in grams of X, Y and Z present at time t seconds after the start of the reaction are x, 10 - x and 20 - x respectively. At any time the rate of formation of X is proportional to the product of the masses of Y and Z present at the time. When t = 0, x =0 and dx/dy = 2.
(i) Show that x and t satisfy the differential equation
Dx/dt = 0.01(10 - x)(20- x)
(ii) Solve this differential equation an obtain an expression for x in terms of t.
(iii) State what happens to the value of x when t becomes large.
Develop a point estimate of the mean value of y when x1 = 180 and x2 = 310 (to 3 decimals).
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