Invasive species often experience exponential population growth when introduced
ID: 2896035 • Letter: I
Question
Invasive species often experience exponential population growth when introduced into a new environment. Zebra mussels are invasive species that have recently made their way into midwestern lakes. In about six months, a zebra mussel population quadruples in size. Suppose 10 zebra mussels are newly introduced into a local lake. Find a function Z(t) that models the population of zebra mussels t years after they have been introduced. Use the function to determine how many zebra mussels there will be in 15 months. Use the function to determine how many months will pass before the lake has one million zebra mussels.
Explanation / Answer
If ‘a’ is the initial population of the Zebra mussels, then every six months, the population of Zebra mussels quadruples, i.e. it becomes 4a. In t years, i.e. in 2t intervals of 6 months each, the population of Zebra mussels can be computed with the help of a geometric series a, ar, ar2, ar3, …. arn-1 ( the series being finite with n terms) where a is the 1st term , r is the common ratio and n is the number of terms in the series..
Here, in 2t years, there will be 2t + 1 terms in the geometric series including the initial term. Thus in the above series, a = 10, r = 4 and n = 2t + 1. Then the population of the Zebra mussels after 2t years is the (2t +1)st term of the geometric series I.e. 10* ( 42t)
In 15 months, t = 15/12 = 1.25. Then the population of Zebra mussels after 15 months will be 10*(42.5 ) = 10 * 25 = 320
If after t years, the population of the Zebra mussels become 1 million , then we have
1000000 = 10 * (42t) or, 100000= 42t or,105 = 42t Taking logarithms of both sides, we have 5 log 10 = 2t log 4 or, t = (5log10)/(2log4) = 5/1.20 years or (5/1.20) * 12 months = 50 months, i.e. 4 years and 2 months.
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