Zebra mussels are an invasive species outside of their native range in eastern Europe. They have recently been introduced in a lake where they are harming the local wildlife. This table shows the predicted number of zebra mussels in the lake over time.

Question

Time (years) Population
1 4,000
2 10,000
3 25,000
4 62,500
Which function describes the predicted number of zebra mussels in the lake after t years?

Answer 1

The function that describes the population of zebra mussels in the lake after t years is an exponential function. The general form of an exponential function is given by
\[ P(t) = P_0 \times (1 + r)^t \]
where:
- P(t) is the population of zebra mussels after t years,
- P0 is the initial population,
- r is the growth rate,
- t is the time in years.

From the table, we can see that the population of zebra mussels is increasing rapidly. To find the growth rate, we can calculate the ratio of the population at t=2 to the population at t=1:
\[ r = \frac{P(2) - P(1)}{P(1)} = \frac{10,000 - 4,000}{4,000} = 1.5 \]
Therefore, the growth rate is 1.5 or 150%.

Using the initial population P0 = 4,000 and the growth rate r = 1.5, the function that predicts the number of zebra mussels in the lake after t years is:
\[ P(t) = 4,000 \times (1 + 1.5)^t \]