The table gives estimates of the world population, in millions, from 1750 to 2000. (a) Use the...

Question:

The table gives estimates of the world population, in millions, from 1750 to 2000.

year population
1750 790
1800 980
1850 1260
1900 1650
1950 2560
2000 6080

(a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950.

(b) Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950.

(c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000.

Population Models using Exponential Growth

Population growth can be modeled by an exponential equation {eq}P(t)=P_0e^{rt} {/eq} where {eq}P_0 {/eq} is the population at time {eq}t=0 {/eq}, {eq}P(t) {/eq} is the population at time {eq}t {/eq}, and {eq}r {/eq} is the continuous annual growth rate. Although population growth can be modeled in this way, the models need to be updated fairly frequently as population growth is not strictly exponential over long periods of time.

Answer and Explanation:

(a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950.


Let {eq}t {/eq} be the number of years since 1750. The population is modeled by {eq}P(t)=790e^{rt} {/eq} and we can substitute the figures for 1800 to find the rate. We have

{eq}980=790e^{r(50)}\\ \dfrac{98}{79}=e^{50r}\\ \ln\left(\dfrac{98}{79}\right) = 50r\\ \dfrac{\ln\left(\frac{98}{79}\right)}{50}=r {/eq}

Substituting this rate into the model and using {eq}t=150 {/eq} and {eq}t=200 {/eq} we have

{eq}\displaystyle P(150)=790e^{\frac{\ln\left(\frac{98}{79}\right)}{50}(150)} \approx 1508 {/eq}

and

{eq}\displaystyle P(200)=790e^{\frac{\ln\left(\frac{98}{79}\right)}{50}(200)} \approx 1871 {/eq}

The population in 1900 will be about 1508 million and in 1950 will be about 1871 million.


(b) Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950.


Let {eq}t {/eq} be the number of years since 1850. The population is modeled by {eq}P(t)=1260e^{rt} {/eq} and we can substitute the figures for 1900 to find the rate. We have

{eq}1650=1260e^{r(50)}\\ \dfrac{55}{42}=e^{50r}\\ \ln\left(\dfrac{55}{42}\right) = 50r\\ \dfrac{\ln\left(\frac{55}{42}\right)}{50}=r {/eq}

Substituting this rate into the model and using {eq}t=100 {/eq} we have

{eq}\displaystyle P(100)=1260e^{\frac{\ln\left(\frac{55}{42}\right)}{50}(100)} \approx 2161 {/eq}

The population in 1950 will be about 2161 million.


(c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000.


Let {eq}t {/eq} be the number of years since 1900. The population is modeled by {eq}P(t)=1650e^{rt} {/eq} and we can substitute the figures for 1950 to find the rate. We have

{eq}2560=1650e^{r(50)}\\ \dfrac{256}{165}=e^{50r}\\ \ln\left(\dfrac{256}{165}\right) = 50r\\ \dfrac{\ln\left(\frac{256}{165}\right)}{50}=r {/eq}

Substituting this rate into the model and using {eq}t=100 {/eq} we have

{eq}\displaystyle P(100)=1650e^{\frac{\ln\left(\frac{256}{165}\right)}{50}(100)} \approx 3972 {/eq}

The population in 2000 will be about 3972 million.


Learn more about this topic:

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Exponential Growth: Definition & Examples

from High School Algebra I: Help and Review

Chapter 6 / Lesson 10
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