A large plantation of area 20 km² is becoming infected with a plant disease. At time t years the area infected is x km² and the rate of increase of x is proportional to the ratio of the area infected to the area not yet infected.

When t = 0, x = 1 and \(\frac{dx}{dt} = 1\).

(a) Show that x and t satisfy the differential equation \(\frac{dx}{dt} = \frac{19x}{20-x}\).

(b) Solve the differential equation and show that when t = 1 the value of x satisfies the equation \(x = e^{0.9 + 0.05x}\).

(c) Use an iterative formula based on the equation in part (b), with an initial value of 2, to determine x correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(d) Calculate the value of t at which the entire plantation becomes infected.