Question

In 1997 the population of New York City was 7.248 million people and in 1998 the population was 7.519 million people. Assume the population of NYC grows exponentially.What is the 1-year growth factor?What is the 2-year growth factor?The population of NYC in 2000 can be determined by______ the population of NYC in 1998 by the _____ -year growth factor.Use your answers from above to help you predict the population of NYC in 2000.What is the 4-year growth factor?

Answer 1

Population of New York City (in millions of people)

1997: 7.248

1998: 7.519

If we assume that the population of NYC grows exponentially, we can write this in functional form as:

`P(y)=A\cdot b^y`

Where A and b are parameters, and y is the number of years passed since 1997. P is the population in millions of people. Using the information above, the population of 1997 can be calculated for y = 0, and in 1998 by y = 1. Then:

`\begin{gathered} P(0)=A\cdot b^0=7.248\Rightarrow A=7.248 \\ P(1)=A\cdot b^1=7.248\cdot b=7.519\Rightarrow b=(7519)/(7248) \end{gathered}`

The expression is:

`P(y)=7.248\cdot((7519)/(7248))^y`

The one-year growth factor is b¹, and the two-year growth factor is b². This is:

`\begin{gathered} 1\text{ year growth factor}\colon b^1=(7519)/(7248)\approx1.037 \\ 2\text{ year growth factor}\colon b^2=((7519)/(7248))^2\approx1.076 \end{gathered}`

The population of NYC in 2000 can be determined by multiplicating the population of NYC in 1998 by the 2 -year growth factor. This is:

`2000\colon7.519\cdot b^2=8.092\text{ millions of people}`

Finally, the 4-year growth factor is b⁴:

`4\text{ year growth factor}\colon b^4=((7519)/(7248))^4\approx1.158`