Textbook Question
Tripling time A quantity increases according to the exponential function y(t) = y₀eᵏᵗ. What is the tripling time for the quantity? What is the time required for the quantity to increase p-fold?
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0. Functions
Exponential Functions
3:06 minutesProblem 7.2.23a
Textbook Question
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.
a. Based on these figures, find the doubling time and the projected population in 2050. Assume the growth rate remains constant.
Verified step by step guidance1
Identify the type of growth model: Since the population grows at a constant percentage rate, this is an example of exponential growth, which can be modeled by the formula \(P(t) = P_0 e^{rt}\), where \(P_0\) is the initial population, \(r\) is the growth rate (as a decimal), and \(t\) is time in years.
Convert the given growth rate from a percentage to a decimal: \(0.79\% = 0.0079\) per year. This will be used as the value of \(r\) in the exponential growth formula.
Calculate the doubling time using the formula for exponential growth doubling time: \(T = \frac{\ln(2)}{r}\). This formula comes from setting \(P(t) = 2P_0\) and solving for \(t\).
Calculate the projected population in 2050 by first determining the time elapsed from 2020 to 2050, which is \(t = 30\) years. Then use the exponential growth formula: \(P(30) = 334.4 \times e^{0.0079 \times 30}\), where 334.4 million is the initial population in 2020.
Evaluate the expression for \(P(30)\) to find the projected population in 2050. This will give the population assuming the growth rate remains constant over the 30 years.
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3mKey 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, often 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 projecting population changes over time assuming a constant growth rate.
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Doubling Time
Doubling time is the period required for a quantity growing exponentially to double in size. It can be calculated using the formula T_d = ln(2)/r, where r is the growth rate expressed as a decimal. This concept helps determine how quickly the population will double under a constant growth rate.
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Natural Logarithm and Its Application
The natural logarithm (ln) is the inverse of the exponential function and is used to solve for time or growth rate in exponential equations. In population projections, ln helps isolate variables when calculating doubling time or future population values, making it a crucial tool for interpreting growth models.
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