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Population of a city was 48,000 in 1990 and 42,000 in 2000. Assuming an exponential model, estimate the population in 2005.

Population of a city was 48,000 in 1990 and 42,000 in 2000. Assuming an exponential model, estimate the population in 2005.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** The population of a city was 48,000 in 1990 and decreased to 42,000 in 2000. Assuming an exponential model, estimate the city's population in 2005. **Analysis:** To solve this problem using an exponential model, we need to determine the rate of change in the population over the given period and then apply that rate to estimate the population for the specified future year. **Approach:** 1. Use the formula for exponential decay, \( P(t) = P_0 \cdot e^{rt} \), where: - \( P(t) \) is the future population, - \( P_0 \) is the initial population, - \( r \) is the rate of growth (negative for decay), - \( t \) is the time in years, - \( e \) is the base of the natural logarithm. 2. Calculate the decay rate \( r \) using the populations in 1990 and 2000. 3. Estimate the population in 2005 using the calculated decay rate. This estimation provides insights into demographic trends and aids in urban planning and resource allocation.
Transcribed Image Text:**Problem Statement:** The population of a city was 48,000 in 1990 and decreased to 42,000 in 2000. Assuming an exponential model, estimate the city's population in 2005. **Analysis:** To solve this problem using an exponential model, we need to determine the rate of change in the population over the given period and then apply that rate to estimate the population for the specified future year. **Approach:** 1. Use the formula for exponential decay, \( P(t) = P_0 \cdot e^{rt} \), where: - \( P(t) \) is the future population, - \( P_0 \) is the initial population, - \( r \) is the rate of growth (negative for decay), - \( t \) is the time in years, - \( e \) is the base of the natural logarithm. 2. Calculate the decay rate \( r \) using the populations in 1990 and 2000. 3. Estimate the population in 2005 using the calculated decay rate. This estimation provides insights into demographic trends and aids in urban planning and resource allocation.
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