/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 Use the exponential growth model... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 33

Use the exponential growth model, \(A=A_{0} e^{k t},\) to show that the time it takes a population to double (to grow from \(A_{0}\) to \(\left.2 A_{0}\right)\) is given by \(t=\frac{\ln 2}{k}\)

Short Answer

Expert verified
The time (t) it takes a population to double in an exponential growth model is given by \(t = \frac{\ln2}{k}\). This is derived from the exponential growth equation by setting the final amount to twice the initial amount, simplifying, and then solving for t.

Step by step solution

01

Understand the Exponential Growth Model

An exponential growth model is represented by the equation \(A=A_0 e^{kt}\), where: \n- \(A\) is the final amount \n- \(A_0\) is the initial amount (i.e., the starting quantity) \n- \(e\) is the base of the natural logarithm (approx. 2.71828), showing that the quantity is growing exponentially \n- \(k\) is the growth rate (constant) \n- \(t\) is the time
02

Apply the Doubling Condition

For a population to double, the final amount \(A\) will be twice the initial amount \(A_0\). Substituting \(2A_0\) for \(A\) in the growth equation results in the following equation: \(2A_0 = A_0 e^{kt}\)
03

Simplify and Solve for t

Divide through the equation by \(A_0\) to get \(2=e^{kt}\). Next, take the natural logarithm (ln) of both sides to get \(\ln 2= kt\). Lastly, rearrange the equation to solve for \(t\), resulting in \(t = \frac{\ln 2}{k}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$ y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2} $$

In Exercises \(128-131,\) graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f\) $$ f(x)=\log x, g(x)=\log (x-2)+1 $$

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ \log (x-15)+\log x=2 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After 100 years, a population whose growth rate is \(3 \%\) will have three times as many people as a population whose growth rate is \(1 \%\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \text { If } \log (x+3)=2, \text { then } e^{2}=x+3 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.