Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.RE.20a

Chapter 7, Problem 7.RE.20a

Population growth The population of a large city grows exponentially with a current population of 1.3 million and a predicted population of 1.45 million 10 years from now.

a. Use an exponential model to estimate the population in 20 years. Assume the annual growth rate is constant.

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Identify the exponential growth model formula: \(P(t) = P_0 \cdot e^{rt}\), where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time in years.

Assign the known values: initial population \(P_0 = 1.3\) million, population after 10 years \(P(10) = 1.45\) million, and \(t = 10\) years.

Use the known values to set up the equation \(1.45 = 1.3 \cdot e^{10r}\) and solve for the growth rate \(r\) by isolating \(r\).

Once \(r\) is found, use the exponential model again to estimate the population after 20 years by calculating \(P(20) = 1.3 \cdot e^{20r}\).

Interpret the result as the estimated population in millions after 20 years, based on the constant annual growth rate.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Growth Model

Exponential growth describes a process where the quantity increases at a rate proportional to its current value, modeled by P(t) = P_0 * e^(rt). Here, P_0 is the initial population, r is the growth rate, and t is time. This model is essential for predicting future population sizes when growth is continuous and constant.

Determining the Growth Rate

To use the exponential model, the growth rate r must be found from known data points. Given initial and future populations at specific times, r can be calculated by rearranging the formula: r = (1/t) * ln(P(t)/P_0). This step is crucial to accurately estimate future populations.

Using the Model for Prediction

Once the growth rate is known, the exponential model can predict population at any future time by substituting t into P(t) = P_0 * e^(rt). This allows estimation of the population after 20 years, assuming the growth rate remains constant over time.

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