After 37 days, the population is three times,
That is,
N37 = 42/14 = 3
Therefore, the relationship between initial population and population after t days, is given as;
N = No*3^(t/37)
Where, N = Population after t days, No = Initial population = 14 beetles, t= number of days.
Therefore,
14,000 = 14*3^(t/37)
1000 = 3^(t/37)
Taking natural logs on both sides,
log 1000 = (t/37) log 3
3 = (t/37) *0.4771
6.2877 = t/37
t = 37*6.2877 = 232.645 days ≈ 233 days
