In a certain chemical reaction the amount, x grams, of a substance is decreasing. The differential equation relating x and t, the time in seconds since the reaction started, is dx/dt = -kx/√t, where k is a positive constant. It is given that x = 100 at the start of the reaction.

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

1. [B1] Separate variables correctly and integrate at least one side
2. [B1] Obtain term ln x
3. [B1] Obtain term - (2/3)kt^(3/2), or equivalent
4. [M1] Evaluate a constant, or use limits x = 100 and t = 0, in a solution containing terms a ln x and bt√t
5. [A1] Obtain correct solution in any form, e.g. ln x = - (2/3)kt√t + ln 100
6. [M1] Substitute x = 80 and t = 25 to form equation in k
7. [M1] Substitute x = 40 and eliminate k
8. [A1] Obtain answer t = 64.1