The population of the world in the year 1650 was about 500 million, and in the year 2010 was 6756 million. (a) Assuming that the population of the world grows exponentially, find the equation for the population P(t) in millions in the year t. (b) Use your answer from part (a) to find the population of the world in the year 1. (c) Is your answer to part (b) reasonable? What does this tell you about how the population of the world grows?

Solution

Verified

Answered 2 years ago

Step 1

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(a) Formula for exponential growth and decay function:

y=y_0e^{kt}

Let $y=P(t)$ represent the population at year $t$ .

\begin{align*} P(t)&=y=y_0e^{kt} \\ P(1650)&=500\text{ million} \\ P(2010)&=6756\text{ million} \end{align*}

Evaluate the formula at $t=1650$ and $t=2010$ :

\begin{align*} 500=P(1650)&=y_0e^{1650k} \\ 6756=P(2010)&=y_0e^{2010k} \end{align*}

When the two equations hold, then the ratio of each side of the two equations is equal as well:

\begin{align*} \frac{6756}{500}&=\frac{y_0e^{2010k}}{y_0e^{1650k}} \\ 13.512&=\frac{e^{2010k}}{e^{1650k}} \\ 13.512&=e^{2010k-1650k} &\color{#4257b2}\frac{a^m}{a^n}=a^{m-n} \\ 13.512&=e^{360k} &\color{#4257b2}\text{Combine like terms} \\ \ln 13.512&=\ln e^{360k} &\color{#4257b2}\text{Take logarithm of both sides} \\ \ln 13.512&=360k &\color{#4257b2}\ln e^x=x \\ \frac{1}{360}\ln 13.512 &=k &\color{#4257b2}\text{Divide each side by 360} \\ 0.007232 &\approx k &\color{#4257b2}\text{Evaluate} \end{align*}

Next, we determine the constant $y_0$ using the previously found equation $500=y_0e^{1650k}$ and $k=0.0072$ :

\begin{align*} y_0&=500e^{-1650k}=500e^{-1650(0.007232 )}=500e^{-11.9328}\approx 0.003285 \end{align*}

Let us use the found value of $k=0.007232$ in the given formula:

P(t)=y_0e^{kt}=0.003285e^{0.007232 t}

Step 2

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(b) Result part (a): $P(t)=0.003285e^{0.007232 t}$

Evaluate the formula for $P(t)$ at $t=1$ :

\begin{align*} P(1)&=0.003285e^{0.007232 (1)} \\ &=0.003285e^{0.007232} &\color{#4257b2}\text{Evaluate product} \\ &\approx 0.003309 &\color{#4257b2}\text{Evaluate} \end{align*}

Thus the population of the world in year 1 was approximately 0.003309 million, or 3309 people.

Step 3

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Since the population of 3309 people is much smaller than the population of 231 million, the answer to part (b) doesn't seem to be reasonable.

This tells us that the population growth doesn't follow an exponential growth pattern.

Result

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(a) $P(t)=0.003285e^{0.007232 t}$

(b) 3309 people

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