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

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

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

Problema 7.2.25

Capitolo 7, Problema 7.2.25

Population of Texas Texas was the third fastest growing state in the United States in 2016. Texas grew from 25.1 million in 2010 to 26.47 million in 2016. Use an exponential growth model to predict the population of Texas in 2025.

Guida verificata passo dopo passo

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

Assign the known values: let \(t=0\) correspond to the year 2010, so \(P_0 = 25.1\) million, and at \(t=6\) (the year 2016), \(P(6) = 26.47\) million.

Use the known populations to find the growth rate \(k\) by substituting into the model: \(26.47 = 25.1 \cdot e^{6k}\). Solve this equation for \(k\) by isolating \(e^{6k}\) and then taking the natural logarithm.

Once \(k\) is found, use the model to predict the population in 2025 by calculating \(P(15) = 25.1 \cdot e^{15k}\), since 2025 is 15 years after 2010.

Interpret the result as the predicted population of Texas in 2025 according to the exponential growth model.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Exponential Growth Model

An exponential growth model describes a quantity that increases at a rate proportional to its current value, often expressed as P(t) = P_0 * e^(rt). Here, P_0 is the initial amount, r is the growth rate, and t is time. This model is used to predict populations or investments growing continuously over time.

Video consigliato:

09:29

Exponential Growth & Decay

Determining the Growth Rate

The growth rate (r) in an exponential model is found by using known data points and solving for r in the equation P(t) = P_0 * e^(rt). By substituting the initial population, later population, and time elapsed, one can isolate r, which quantifies how fast the population grows per unit time.

Video consigliato:

Percorso guidato

04:16

Intro To Related Rates

Using the Model to Make Predictions

Once the growth rate is known, the exponential model can predict future values by plugging in the desired time t. For example, to predict the population in 2025, substitute t as the number of years since the initial time (2010) into the model, allowing estimation of population size at that future date.

Video consigliato:

Percorso guidato

10:17

Using The Velocity Function

Pratica correlata

Domanda del libro di testo

Express sinh⁻¹ x in terms of logarithms.

169

views

Domanda del libro di testo

Inverse identity Show that cosh⁻¹(cosh x) = |x| by using the formula cosh⁻¹ t = ln (t + √(t² – 1)) and considering the cases x ≥ 0 and x < 0.

108

views

Domanda del libro di testo

What is the inverse function of ln x, and what are its domain and range?

262

views

Domanda del libro di testo

37–56. Integrals Evaluate each integral.

∫ sinh x / (1 + cosh x) dx

views

Domanda del libro di testo

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫ 3^{-2x} dx

views

Domanda del libro di testo

13–14. Absolute and relative growth rates Two functions f and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant.

f(t) = 100 + 10.5t, g(t) = 100e^(t/10)

views