The U.S. population is approximated by the function P(t)=(616.5)/(1+4.02 e^-0.5 t) where P(t) is

Name: Problem 26, Chapter 5: Applied Calculus for the Managerial, Life, and Social Sciences
Uploaded: 2021-07-29T06:21:39-08:00
Duration: 1 min 6 s
Channel: Carson Merrill
Description: The U.S. population is approximated by the function P(t)=(616.5)/(1+4.02 e^-0.5 t) where P(t) is measured in millions of people and t is measured in 30 -year intervals, with t=0 corresponding to 1930 . What is the expected population of the United States in 2020(t=3) ?

step 1 For the given problem, we want to consider this population growth model, p of t equals 616 .5, d

Question

The U.S. population is approximated by the function $$P(t)=\frac{616.5}{1+4.02 e^{-0.5 t}}$$ where $P(t)$ is measured in millions of people and $t$ is measured in 30 -year intervals, with $t=0$ corresponding to $1930 .$ What is the expected population of the United States in $2020(t=3) ?$

   The U.S. population is approximated by the function $$P(t)=\frac{616.5}{1+4.02 e^{-0.5 t}}$$ where $P(t)$ is measured in millions of people and $t$ is measured in 30 -year intervals, with $t=0$ corresponding to $1930 .$ What is the expected population of the United States in $2020(t=3) ?$

Applied Calculus for the Managerial, Life, and Social Sciences

Soo T. Tan 10th Edition

Chapter 5, Problem 26

Instant Answer

Solved by Expert Carson Merrill

07/29/2021

Step 1

The function $$P(t)=\frac{616.5}{1+4.02 e^{-0.5 t}}$$ is a model for the population of the United States, where $P(t)$ is the population in millions and $t$ is the time in 30-year intervals from 1930. Show more…

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The U.S. population is approximated by the function $$P(t)=\frac{616.5}{1+4.02 e^{-0.5 t}}$$ where $P(t)$ is measured in millions of people and $t$ is measured in 30 -year intervals, with $t=0$ corresponding to $1930 .$ What is the expected population of the United States in $2020(t=3) ?$

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Key Concepts

Logistic Growth Model

A logistic growth model is used to describe populations that initially grow exponentially and then slow in growth as they approach a maximum limit, known as the carrying capacity. This sigmoidal model is characterized by a rapid growth phase followed by a plateau, reflecting constraints on resources or space.

Exponential Functions

Exponential functions are used to model processes where the rate of change is proportional to the current value. In population models, exponential terms represent growth or decay dynamics, showing how populations increase rapidly under certain conditions before leveling off due to limiting factors.

Function Evaluation

Function evaluation involves substituting a specific input value into a mathematical function to determine the corresponding output. This process is crucial in applied scenarios such as predicting population sizes at a given time by inserting the time value into the model.

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