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 dx/dt = 1.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [M1] State or imply equation of the form dx/dt = k * x/(20−x)
2. [A1 AG] Obtain k = 19
3. [M1] Separate variables and integrate at least one side
4. [A1 A1] Obtain terms 20 ln x − x and 19t, or equivalent
5. [M1] Evaluate a constant or use t = 0 and x = 1 as limits in a solution containing terms a ln x and bt
6. [A1 AG] Substitute t = 1 and rearrange the equation in the given form
7. [M1] Use xₙ₊₁ = e^(0.9+0.05xₙ) correctly at least once
8. [A1] Obtain final answer x = 2.83
9. [A1] Show sufficient iterations to 4 decimal places to justify 2.83 to 2 d.p. or show there is a sign change in the interval (2.825, 2.835)
10. [B1] Set x = 20 and obtain answer t = 2.15