We appreciate your visit to Suppose the population of a country in 1985 was 145 million In 1995 it was 190 million What is the population of the country in. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
We start by assuming that the population grows continuously according to the model
[tex]$$
P(t) = P_0 \, e^{rt},
$$[/tex]
where [tex]$P_0$[/tex] is the initial population, [tex]$r$[/tex] is the continuous growth rate, and [tex]$t$[/tex] is the number of years.
1. Determine the Growth Rate
The population in 1985 is given as [tex]$145$[/tex] million and in 1995 as [tex]$190$[/tex] million. Since the time between 1985 and 1995 is [tex]$10$[/tex] years, we can set up the equation
[tex]$$
190 = 145 \, e^{r \cdot 10}.
$$[/tex]
To solve for [tex]$r$[/tex], divide both sides by [tex]$145$[/tex]:
[tex]$$
\frac{190}{145} = e^{10r}.
$$[/tex]
Taking the natural logarithm of both sides gives
[tex]$$
\ln\left(\frac{190}{145}\right) = 10r.
$$[/tex]
Therefore, the growth rate [tex]$r$[/tex] is
[tex]$$
r = \frac{1}{10} \ln\left(\frac{190}{145}\right).
$$[/tex]
Calculating the value yields approximately
[tex]$$
r \approx 0.027 \quad \text{per year}.
$$[/tex]
2. Calculate the Population in 2005
The year 2005 is [tex]$10$[/tex] years after 1995. Using the population from 1995 ([tex]$190$[/tex] million) as the starting point, we calculate the population in 2005 by
[tex]$$
P(2005) = 190 \, e^{r \cdot 10}.
$$[/tex]
Substituting the approximate value for [tex]$r$[/tex], we get
[tex]$$
P(2005) = 190 \, e^{0.027 \cdot 10}.
$$[/tex]
Evaluating this expression gives an approximate population of about [tex]$248.97$[/tex] million.
3. Matching with the Options
Among the choices provided:
- Option A: [tex]$P=145 e^{(0.027)(10)}$[/tex]
- Option B: [tex]$P=145 e^{(0.027)(20)}$[/tex]
- Option C: [tex]$P=190 e^{(0.027)(10)}$[/tex]
- Option D: [tex]$P=190 e^{(0.027)(20)}$[/tex]
The correct expression that we derived is
[tex]$$
\boxed{190 \, e^{(0.027)(10)}},
$$[/tex]
which corresponds to Option C.
Thus, the correct answer is Option C.
[tex]$$
P(t) = P_0 \, e^{rt},
$$[/tex]
where [tex]$P_0$[/tex] is the initial population, [tex]$r$[/tex] is the continuous growth rate, and [tex]$t$[/tex] is the number of years.
1. Determine the Growth Rate
The population in 1985 is given as [tex]$145$[/tex] million and in 1995 as [tex]$190$[/tex] million. Since the time between 1985 and 1995 is [tex]$10$[/tex] years, we can set up the equation
[tex]$$
190 = 145 \, e^{r \cdot 10}.
$$[/tex]
To solve for [tex]$r$[/tex], divide both sides by [tex]$145$[/tex]:
[tex]$$
\frac{190}{145} = e^{10r}.
$$[/tex]
Taking the natural logarithm of both sides gives
[tex]$$
\ln\left(\frac{190}{145}\right) = 10r.
$$[/tex]
Therefore, the growth rate [tex]$r$[/tex] is
[tex]$$
r = \frac{1}{10} \ln\left(\frac{190}{145}\right).
$$[/tex]
Calculating the value yields approximately
[tex]$$
r \approx 0.027 \quad \text{per year}.
$$[/tex]
2. Calculate the Population in 2005
The year 2005 is [tex]$10$[/tex] years after 1995. Using the population from 1995 ([tex]$190$[/tex] million) as the starting point, we calculate the population in 2005 by
[tex]$$
P(2005) = 190 \, e^{r \cdot 10}.
$$[/tex]
Substituting the approximate value for [tex]$r$[/tex], we get
[tex]$$
P(2005) = 190 \, e^{0.027 \cdot 10}.
$$[/tex]
Evaluating this expression gives an approximate population of about [tex]$248.97$[/tex] million.
3. Matching with the Options
Among the choices provided:
- Option A: [tex]$P=145 e^{(0.027)(10)}$[/tex]
- Option B: [tex]$P=145 e^{(0.027)(20)}$[/tex]
- Option C: [tex]$P=190 e^{(0.027)(10)}$[/tex]
- Option D: [tex]$P=190 e^{(0.027)(20)}$[/tex]
The correct expression that we derived is
[tex]$$
\boxed{190 \, e^{(0.027)(10)}},
$$[/tex]
which corresponds to Option C.
Thus, the correct answer is Option C.
Thanks for taking the time to read Suppose the population of a country in 1985 was 145 million In 1995 it was 190 million What is the population of the country in. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
