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 7.2.23b

Chapter 7, Problem 7.2.23b

Projection sensitivity
According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr.
b. Suppose the actual growth rate is instead 0.7%. What are the resulting doubling time and projected 2050 population?

Verified step by step guidance

Identify the formula for exponential growth of a population: $P(t) = P_0 \times e^{rt}$, where $P_0$ is the initial population, $r$ is the growth rate (expressed as a decimal), and $t$ is the time in years from the initial time.

Calculate the doubling time using the formula for exponential growth doubling time: $T_d = \frac{\ln(2)}{r}$. Here, $r$ is the actual growth rate of 0.7%, so convert it to decimal form as $r = 0.007$.

Determine the time interval from the base year to 2050. Since the base year is 2020, the time $t$ is \$2050 - 2020 = 30$ years.

Use the exponential growth formula to find the projected population in 2050: $P(30) = 334.4 \times e^{0.007 \times 30}$. Here, 334.4 million is the population in 2020, and $r = 0.007$ is the growth rate.

Evaluate the expressions for doubling time and projected population after substituting the values, but do not calculate the final numerical results as per instructions.

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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 populations increasing at a rate proportional to their current size, modeled by P(t) = P_0 * e^(rt), where P_0 is the initial population, r is the growth rate, and t is time. This model helps predict future population sizes based on continuous growth rates.

Doubling Time

Doubling time is the period required for a quantity growing exponentially to double in size. It is calculated using the formula T_d = ln(2)/r, where r is the growth rate. This concept helps understand how quickly a population grows under a given rate.

Applying Growth Rate to Population Projections

To project future population, the exponential growth formula uses the actual growth rate and time elapsed. Adjusting the growth rate changes the projected population, illustrating sensitivity in predictions and the importance of accurate growth estimates.

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