# 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}

Question
Exponential models
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: $$A=1094.2e^{0.016t}$$

2021-02-07
Given:
The exponential model that describe the population of different countries:
Country 1: $$A=26.3e^{0.029t}$$
Country 2: $$A=127.7e^{0.007t}$$
Country 3: $$A=148.5e^{-0.0091t}$$
Country 4: $$A=1094.2e^{0.016t}$$
Population growth is equal to differentiation of exponential model with respect to ‘t’,
So now differentiating the given function one by one:
$$\frac{d}{dt}(A_1)=\frac{d}{dt}(26.3e^{0.029t})$$
$$\frac{dA_1}{dt}=(26.3\times0.029e^{0.029t})$$
$$\frac{dA_1}{dt}=(0.7627e^{0.029t}$$
$$\frac{d}{dt}(A_2)=\frac{d}{dt}(127.7e^{0.007t})$$
$$\frac{dA_1}{dt}=(127.7\times0.007e^{0.007t}$$
$$\frac{dA_1}{dt}=(0.8939e^{0.007t})$$
$$\frac{d}{dt}(A_3)=\frac{d}{dt}(148.5e^{-0.009t})$$
$$\frac{dA_3}{dt}=\frac{d}{dt}(148.5\times(-0.009)e^{-0.009t})$$
$$\frac{dA_3}{dt}=-(1.3365e^{-0.009t})$$
$$\frac{d}{dt}(A_4)=\frac{d}{dt}(1094.2e^{0.016t}$$
$$\frac{dA_4}{dt}=\frac{d}{dt}(1094.2\times0.016e^{0.016t})$$
$$\frac{dA_4}{dt}=(17.5072e^{0.016t})$$
To check which one is larger, by comparing the quantities of growth rate it can be concluded that:
Country 4: $$A_4=1094.2e^{0.016t}$$
Has the greates growth rate.
To find the percentage change in population:
Country 4: $$A_4=1092.2e^{0.016t}$$
Now put t=0,
$$A_4=1094.2e^{0.016\times0}$$
$$A_4=1094.2e^0$$
$$A_4=1094.2$$
Now put $$t=1,$$
$$A_4=1094.2e^{0.016\times1}$$
$$A_4=1094.2e^{0.016}$$
$$A_4=1111.8$$
Now,
Percentage change $$=\frac{1111.8-1094.2}{1094.2}\times100$$
Percentage change $$=1.60%$$
Country 4: $$A=1094.2e^{0.016t}$$
Has the greates growth rate.
Percentage change in population each year $$=1.60%$$

### Relevant Questions

The exponential models describe the population of the indicated country, A, in millions, t years after 2010.Which country has the greatest growth rate? By what percentage is the population of that country increasing each year?
India, $$A=1173.1e^(0.008t)$$
Iraq, $$A=31.5e^(0.019t)$$
Japan, $$A=127.3e^(0.006t)$$
Russia, $$A=141.9e^(0.005t)$$
The exponential models describe the population of the indicated country, A, in millions, t years after 2010. Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year?
Country B $$A=1193.1e^{0.006t}$$
Country C $$A=36.5e^{0.017t}$$
Country D $$A=121.7e^{-0.005t}$$
Country E $$A=145.3e^{-0.004t}$$
The exponential models describe the population of the indicated country, A, in millions, t years after 2010. Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year?
Country B $$\displaystyle{A}={1193.1}{e}^{{{0.006}{t}}}$$
Country C $$\displaystyle{A}={36.5}{e}^{{{0.017}{t}}}$$
Country D $$\displaystyle{A}={121.7}{e}^{{-{0.005}{t}}}$$
Country E $$\displaystyle{A}={145.3}{e}^{{-{0.004}{t}}}$$
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.
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.
Often new technology spreads exponentially. Between 1995 and 2005, each year the number of Internet domain hosts was 1.43 times the number of hosts in the preceding year. In 1995, the number of hosts was 8.2 million.
(a) Explain why the number of hosts is an exponential function of time. The number of hosts grows by a factor of -----? each year, this is an exponential function because the number is growing by ------? decreasing constant increasing multiples.
(b) Find a formula for the exponential function that gives the number N of hosts, in millions, as a function of the time t in years since 1995.
(c) According to this model, in what year did the number of hosts reach 49 million?
Use a calculator with a $$y^x$$ key or a key to solve: India is currently one of the world’s fastest-growing countries. By 2040, the population of India will be larger than the population of China, by 2050, nearly one-third of the world’s population will live in these two countries alone. The exponential function $$f(x)=574(1.026)^x$$ models the population of India, f(x), in millions, x years after 1974.