Question

1. the number of burmese pythons in the florida everglades has been increasing exponentially since the 1970s when irresponsible pet owners released them into the ecosystem. in 2019 it was estimated that there were 200,000. in 2020 it was estimated that there were 252,000.

a) write an equation in the form ( y = acdot b^{x} ) to model the population of burmese python in the florida everglades, where ( x ) is the number of years since 1975. round ( b ) to the nearest tenth.
b) interpret the value of ( a ) for this model.

Explanation:

Step1: Determine the value of \(x\) and \(y\) for the first data point

In 1975 (\(x = 0\)), assume the initial population. But in 2019 (\(x=2019 - 1975=44\)), \(y = 252000\), and if we assume in 1975 (\(x = 0\)) the population \(y=a\). Wait, no, actually, the general form is \(y=a\cdot b^{x}\). When \(x = 0\) (1975), if we consider the start - point (assuming the growth starts from 1975), and in 2019 (\(x = 44\)), \(y=252000\). But wait, no, hold on. Wait, the formula \(y=a\cdot b^{x}\). When \(x = 0\) (1975), if we assume the first data - point. Wait, no, actually, the problem says "since the 1970s when irresponsible pet owners released them into the ecosystem". Let's assume \(x = 0\) corresponds to 1975 (mid - 1970s). And in 2019 (\(x=2019 - 1975 = 44\)), \(y = 252000\). Also, if we assume that in 1975 (\(x = 0\)), the initial population \(y=a\). But wait, no, the problem might have a typo. Wait, no, the formula \(y=a\cdot b^{x}\). When \(x = 0\), \(y=a\). So \(a\) is the initial population of Burmese pythons in the Florida Everglades in 1975.

Step2: Calculate \(b\)

We know that \(y=a\cdot b^{x}\). Substituting \(x = 44\) and \(y = 252000\) (assuming \(a = 200000\) (in 1975, \(x = 0\), \(y=a\), and in 2019 (\(x = 44\)), \(y = 252000\))

$$252000=200000\cdot b^{44}$$

$$b^{44}=\frac{252000}{200000}=1.26$$

$$b = 1.26^{\frac{1}{44}}$$

Using the formula \(a^{m/n}=\sqrt[n]{a^{m}}\), we have \(b=\sqrt[44]{1.26}\)

$$b\approx1.005$$

Step3: Write the equation

The equation is \(y = 200000\cdot(1.005)^{x}\)

Answer:

a) \(y = 200000\cdot(1.005)^{x}\)
b) The value of \(a = 200000\) represents the initial number (in 1975) of Burmese pythons in the Florida Everglades ecosystem.