# If the population had continued growing at its 1920 growth rate for the rest of the century, what...

## Question:

If the population had continued growing at its 1920 growth rate for the rest of the century, what would the population have been at the start of the year 2000? (Give your answer in millions of people, correct to one decimal place.)

The grow rate is 0.462

## Population Growth:

An initial population of {eq}x_0 {/eq} growing or compounding at a constant yearly rate {eq}r {/eq} grows after {eq}t {/eq} years to the number {eq}x {/eq} calculated according to the equation {eq}x=x_0e^{rt} {/eq}.

## Answer and Explanation:

Denote the population at the start of the year {eq}1920 {/eq} as {eq}P_0 {/eq}. Counting from 1920, 2000 is year number {eq}t=2000-1920=80 {/eq}.

It is to be assumed that the yearly growth rate for the population remained constant at {eq}r=0.462 {/eq}

The population {eq}P {/eq} after {eq}t=80 {/eq} years is given by the equation

{eq}\text{Final population }=\text{ Initial population }* e^{rt} {/eq}

So

{eq}\begin{align*} P&=P_0e^{0.462*80} \\[2ex] \Rightarrow P&\approx P_0*1.126*10^{16} \end{align*} {/eq}

Answer: The population at the start of the year 2000 would have been approximately {eq}1.126*10^{16} {/eq} times the population at the start of the year 1920.

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from Math 104: Calculus

Chapter 15 / Lesson 3