Question

Over time, the number of organisms in a population increases exponentially. The table below shows the approximate number of organisms after y years.

y years
number of organisms, n
1
55
2
60
3
67
4
75

The environment in which the organism lives can support at most 600 organisms. Assuming the trend continues, after how many years will the environment no longer be able to support the population?
12
24
61
82

Answer 1

Let n(y) = A r^y. From y = 1 and y = 4:
r = (n4 / n1)^(1/(4-1)) = (75/55)^(1/3) ≈ 1.10888.
Since n(1)=55, n(y)=55·r^(y-1). Solve 55·r^(y-1)=600:
y − 1 = ln(600/55) / ln r ≈ ln(10.9091)/ln(1.10888) ≈ 23.11,
so y ≈ 24.11 years.

Thus the population exceeds 600 after about 24 years. Answer: 24.