The population of a small town was 3,600 people in the year 2015. The population increases by 4.5% every year. Write the exponential equation that models the population of the town t years after 2015. Then use your equation to estimate the population in the year 2025?

Question
Exponential models
asked 2021-02-09
The population of a small town was 3,600 people in the year 2015. The population increases by 4.5% every year.
Write the exponential equation that models the population of the town t years after 2015.
Then use your equation to estimate the population in the year 2025?

Answers (1)

2021-02-10
The general form of exponential growth is
\(\displaystyle{y}={y}_{{0}}{e}^{{{k}{t}}}\)
where,
\(\displaystyle{y}_{{0}}=\) initial population
k= constant growth rate
In this question, we have been given the initial population and a constant growth rate which is
\(\displaystyle{y}_{{0}}={3600}\)
k=4.5%=0.045
Substituting the values in equation (1) we have the equation that models the population of the town t years after 2015
\(\displaystyle{y}={3600}{e}^{{{0.045}{t}}}\)
Here we have to calculate the population in 2025 which means the time t = 10 after 2015
\(\displaystyle{y}={3600}{e}^{{{0.045}{\left({10}\right)}}}\)
\(\displaystyle{y}={3600}{e}^{{{0.45}}}\)
\(\displaystyle{y}={3600}\times{1.56}\)
\(\displaystyle{y}={5647.49}\)
\(\displaystyle{y}={5648}\)(approx)
hence, the population in 2025 is y=5648
0

Relevant Questions

asked 2020-11-23
The population of a small town was 3,600 people in the year 2015. The population increases by 4.5% every year.
Write the exponential equation that models the population of the town t years after 2015.
Then use your equation to estimate the population in the year 2025?
asked 2021-02-11
Several models have been proposed to explain the diversification of life during geological periods. According to Benton (1997), The diversification of marine families in the past 600 million years (Myr) appears to have followed two or three logistic curves, with equilibrium levels that lasted for up to 200 Myr. In contrast, continental organisms clearly show an exponential pattern of diversification, and although it is not clear whether the empirical diversification patterns are real or are artifacts of a poor fossil record, the latter explanation seems unlikely. In this problem, we will investigate three models fordiversification. They are analogous to models for populationgrowth, however, the quantities involved have a differentinterpretation. We denote by N(t) the diversification function,which counts the number of taxa as a function of time, and by rthe intrinsic rate of diversification.
(a) (Exponential Model) This model is described by \(\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{e}}}{N}\ {\left({8.86}\right)}.\) Solve (8.86) with the initial condition N(0) at time 0, and show that \(\displaystyle{r}_{{{e}}}\) can be estimated from \(\displaystyle{r}_{{{e}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ {\left({8.87}\right)}\)
(b) (Logistic Growth) This model is described by \(\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{l}}}{N}\ {\left({1}\ -\ {\frac{{{N}}}{{{K}}}}\right)}\ {\left({8.88}\right)}\) where K is the equilibrium value. Solve (8.88) with the initial condition N(0) at time 0, and show that \(\displaystyle{r}_{{{l}}}\) can be estimated from \(\displaystyle{r}_{{{l}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{K}\ -\ {N}{\left({0}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ +\ {\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{K}\ -\ {N}{\left({t}\right)}}}}\right]}\ {\left({8.89}\right)}\) for \(\displaystyle{N}{\left({t}\right)}\ {<}\ {K}.\)
(c) Assume that \(\displaystyle{N}{\left({0}\right)}={1}\) and \(\displaystyle{N}{\left({10}\right)}={1000}.\) Estimate \(\displaystyle{r}_{{{e}}}\) and \(\displaystyle{r}_{{{l}}}\) for both \(\displaystyle{K}={1001}\) and \(\displaystyle{K}={10000}.\)
(d) Use your answer in (c) to explain the following quote from Stanley (1979): There must be a general tendency for calculated values of \(\displaystyle{\left[{r}\right]}\) to represent underestimates of exponential rates,because some radiation will have followed distinctly sigmoid paths during the interval evaluated.
(e) Explain why the exponential model is a good approximation to the logistic model when \(\displaystyle\frac{{N}}{{K}}\) is small compared with 1.
asked 2020-10-28
The fox population in a certain region has a continuous growth rate of 6 percent per year. It is estimated that the population in the year 2000 was 18900.
(a) Find a function that models the population t years after 2000 (t=0 for 2000). Hint: Use an exponential function with base e.
(b) Use the function from part (a) to estimate the fox population in the year 2008.
asked 2021-01-05
The fox population in a certain region has a continuous growth rate of 6 percent per year. It is estimated that the population in the year 2000 was 18900.
(a) Find a function that models the population t years after 2000 (t=0 for 2000). Hint: Use an exponential function with base e.
(b) Use the function from part (a) to estimate the fox population in the year 2008.
asked 2020-11-08
The burial cloth of an Egyptian mummy is estimated to contain 560 g of the radioactive materialcarbon-14, which has a half life of 5730 years.
a. Complete the table below. Make sure you justify your answer by showing all the steps.
\(\begin{array}{|l|l|l|}\hline t(\text{in years})&m(\text{amoun of radioactive material})\\\hline0&\\\hline5730\\\hline11460\\\hline17190\\\hline\end{array}\)
b. Find an exponential function that models the amount of carbon-14 in the cloth, y, after t years. Make sure you justify your answer by showing all the steps.
c. If the burial cloth is estimated to contain 49.5% of the original amount of carbon-14, how long ago was the mummy buried. Give exact answer. Make sure you justify your answer by showing all the steps.
asked 2021-02-12
The burial cloth of an Egyptian mummy is estimated to contain 560 g of the radioactive materialcarbon-14, which has a half life of 5730 years.
a. Complete the table below. Make sure you justify your answer by showing all the steps.
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{t}{\left(\text{in years}\right)}&{m}{\left(\text{amoun of radioactive material}\right)}\backslash{h}{l}\in{e}{0}&\backslash{h}{l}\in{e}{5730}\backslash{h}{l}\in{e}{11460}\backslash{h}{l}\in{e}{17190}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
b. Find an exponential function that models the amount of carbon-14 in the cloth, y, after t years. Make sure you justify your answer by showing all the steps.
c. If the burial cloth is estimated to contain 49.5% of the original amount of carbon-14, how long ago was the mummy buried. Give exact answer. Make sure you justify your answer by showing all the steps.
asked 2021-02-06
The exponential models the population of the indicated country, A, in millions, t years after 20006. Which country has the greatest growth rate? By what percentage is the population of that country increasing each year?
Country 1: \(A=26.3e^{0.029t}\)
Country 2: \(A=127.7e^{0.007t}\)
Country 3: \(A=148.5e^{-0.0091t}\)
Country 4: \(A=1094.2e^{0.016t}\)
asked 2021-02-21
Use the exponential growth model, \(A=A_0e^{kt}\). In 1975, the population of Europe was 679 million. By 2015, the population had grown to 746 million.
Solve,
a. Find an exponential growth function that models the data for 1975 through 2015.
b. By which year, to the nearest year, will the European population reach 800 million?
asked 2021-02-11
A certain region had an initial rabbit population of 24, and the population is increasing at a rate of 26% each year due to a lack of natural predators. Write an exponential function that models N, the number of rabbits after t years, and use it to find the expected number of rabbits after 50 years.
asked 2021-01-08
A certain region had an initial rabbit population of 24, and the population is increasing at a rate of 26% each year due to a lack of natural predators. Write an exponential function that models N, the number of rabbits after t years, and use it to find the expected number of rabbits after 50 years.
...