Suppose a certain country's population has constant relative birth and death rates of 97 births per thousand people per year and 47 deaths per thousand people per year respectively. Assume also that approximately 30,000 people emigrate (leave) from the country every year. Which equation best models the population P = P(t) of the country, where t is in years? dP A. dt = 50P + 30000 dP B. at C. D. **** dP at dP 0.05P - 30000 = 0.05P+ 30000 dt = 0.5P 30000 dp E. dt = 0.5P - 30
Suppose a certain country's population has constant relative birth and death rates of 97 births per thousand people per year and 47 deaths per thousand people per year respectively. Assume also that approximately 30,000 people emigrate (leave) from the country every year. Which equation best models the population P = P(t) of the country, where t is in years? dP A. dt = 50P + 30000 dP B. at C. D. **** dP at dP 0.05P - 30000 = 0.05P+ 30000 dt = 0.5P 30000 dp E. dt = 0.5P - 30
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 16EQ
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![**Differential Equation Model of Population Change**
Consider a scenario in which a country's population exhibits stable birth and death rates, with 97 births per thousand people annually and 47 deaths per thousand people annually. Additionally, it is assumed that approximately 30,000 individuals emigrate (leave) from the country each year. We aim to determine which differential equation effectively models the population \( P(t) \) of the country as a function of time \( t \), in years.
The potential differential equations provided for this scenario are:
- A. \( \frac{dP}{dt} = 50P + 30000 \)
- B. \( \frac{dP}{dt} = 0.05P - 30000 \)
- C. \( \frac{dP}{dt} = 0.05P + 30000 \)
- D. \( \frac{dP}{dt} = 0.5P - 30000 \)
- E. \( \frac{dP}{dt} = 0.5P - 30 \)
**Analysis of Births and Deaths**
Given:
- Birth rate: \( 97 \) births per 1000 people per year
- Death rate: \( 47 \) deaths per 1000 people per year
Calculating net growth rate (births minus deaths):
\[ \text{Net growth rate} = 97 - 47 = 50 \text{ per 1000 people per year} \]
\[ 50 \text{ per 1000 people per year} = 0.05 \text{ or } 5\% \text{ per year} \]
**Incorporating Emigration**
Since \( 30,000 \) people emigrate each year, this value needs to be subtracted from the population growth term.
The resulting differential equation should include:
- A growth term proportional to the current population ( \( 0.05P \))
- A subtraction term accounting for emigration ( \( -30000 \))
**Conclusion**
Thus, the differential equation that best models the population of the country is:
\[ \boxed{\text{B. } \frac{dP}{dt} = 0.05P - 30000} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6bed085c-3ecb-47a6-9147-a081ba32700e%2Fae9e9d83-c151-432a-9c8a-1e657c76556d%2Fm6qu40d_processed.png&w=3840&q=75)
Transcribed Image Text:**Differential Equation Model of Population Change**
Consider a scenario in which a country's population exhibits stable birth and death rates, with 97 births per thousand people annually and 47 deaths per thousand people annually. Additionally, it is assumed that approximately 30,000 individuals emigrate (leave) from the country each year. We aim to determine which differential equation effectively models the population \( P(t) \) of the country as a function of time \( t \), in years.
The potential differential equations provided for this scenario are:
- A. \( \frac{dP}{dt} = 50P + 30000 \)
- B. \( \frac{dP}{dt} = 0.05P - 30000 \)
- C. \( \frac{dP}{dt} = 0.05P + 30000 \)
- D. \( \frac{dP}{dt} = 0.5P - 30000 \)
- E. \( \frac{dP}{dt} = 0.5P - 30 \)
**Analysis of Births and Deaths**
Given:
- Birth rate: \( 97 \) births per 1000 people per year
- Death rate: \( 47 \) deaths per 1000 people per year
Calculating net growth rate (births minus deaths):
\[ \text{Net growth rate} = 97 - 47 = 50 \text{ per 1000 people per year} \]
\[ 50 \text{ per 1000 people per year} = 0.05 \text{ or } 5\% \text{ per year} \]
**Incorporating Emigration**
Since \( 30,000 \) people emigrate each year, this value needs to be subtracted from the population growth term.
The resulting differential equation should include:
- A growth term proportional to the current population ( \( 0.05P \))
- A subtraction term accounting for emigration ( \( -30000 \))
**Conclusion**
Thus, the differential equation that best models the population of the country is:
\[ \boxed{\text{B. } \frac{dP}{dt} = 0.05P - 30000} \]
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