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Using a Logistic Equation In Exercises 11-14, the logistic equation models the growth of a population. Use the equation to (a) find the value of k , (b) find the carrying capacity, (c) find the initial population, (d) determine when the population will reach 50% of its carrying capacity, and (e) write a logistic differential equation that has the solution P(t). P ( t ) = 6000 1 + 4999 e − 0.8 t

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Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285774770

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Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285774770
Chapter 6.4, Problem 11E
Textbook Problem
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Using a Logistic Equation In Exercises 11-14, the logistic equation models the growth of a population. Use the equation to (a) find the value of k , (b) find the carrying capacity, (c) find the initial population, (d) determine when the population will reach 50% of its carrying capacity, and (e) write a logistic differential equation that has the solution P(t).

P ( t ) = 6000 1 + 4999 e 0.8 t

(a)

To determine

To calculate: The value of k of the logistic equation P(t)=60001+4999e0.8t.

Explanation of Solution

Given:

The logistic equation P(t)=60001+4999e0.8t.

Formula used:

The general logistic equation is y=L1+bekt

(b)

To determine

To calculate: The value of carrying capacity of the logistic equation P(t)=60001+4999e0.8t.

(c)

To determine

To calculate: The value of initial population of the logistic equation P(t)=60001+4999e0.8t.

(d)

To determine

To calculate: The time when the population will be 50% of its carrying capacity of the logistic equation P(t)=60001+4999e0.8t.

(e)

To determine

To calculate: The logistic differential equation of the logistic equation P(t)=60001+4999e0.8t.

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Chapter 6 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)
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In Exercises 65 and 66, determine...Ch. 6.5 - True or False? In Exercises 65 and 66, determine...Ch. 6.6 - Analyzing Predator-Prey Equations In Exercises 3...Ch. 6.6 - Analyzing Predator-Prey EquationsIn Exercises 14,...Ch. 6.6 - Analyzing Predator-Prey EquationsIn Exercises 14,...Ch. 6.6 - Analyzing Predator-Prey Equations In Exercises 3...Ch. 6.6 - Predator-Prey Equations and Slope Fields In...Ch. 6.6 - Predator-Prey Equations and Slope Fields In...Ch. 6.6 - Predator-Prey Equations and Slope Fields In...Ch. 6.6 - Predator-Prey Equations and Slope Fields In...Ch. 6.6 - Rabbits and Foxes In Exercises 9-12, consider a...Ch. 6.6 - Rabbits and Foxes In Exercises 9-12, consider a...Ch. 6.6 - Rabbits and Foxes In Exercises 9-12, consider a...Ch. 6.6 - Rabbits and Foxes In Exercises 9-12, consider a...Ch. 6.6 - Prairie Dogs and Black-Footed Ferrets In Exercises...Ch. 6.6 - Prairie Dogs and Black-Footed Ferrets In Exercises...Ch. 6.6 - Prairie Dogs and Black-Footed Ferrets In Exercises...Ch. 6.6 - Prairie Dogs and Black-Footed Ferrets In Exercises...Ch. 6.6 - Critical Point as the Initial Condition In...Ch. 6.6 - Critical Point as the Initial Condition In...Ch. 6.6 - Analyzing Competing-Species Equations In Exercises...Ch. 6.6 - Analyzing Competing-Species Equations In Exercises...Ch. 6.6 - Analyzing Competing-Species Equations In Exercises...Ch. 6.6 - Analyzing Competing-Species Equations In Exercises...Ch. 6.6 - Bass and Trout In Exercises 23 and 24, consider a...Ch. 6.6 - Bass and Trout In Exercises 23 and 24, consider a...Ch. 6.6 - Bass and Trout In Exercises 25 and 26, consider a...Ch. 6.6 - Bass and Trout In Exercises 25 and 26, consider a...Ch. 6.6 - Critical Point as the Initial Condition In...Ch. 6.6 - Critical Point as the Initial Condition In...Ch. 6.6 - WRITING ABOUT CONCEPTS Separation of Variables Are...Ch. 6.6 - EXPLORING CONCEPTS Determining Initial Values...Ch. 6.6 - Determining Initial Values Given a set of compete...Ch. 6.6 - HOW DO YOU SEE IT? The populations of two species...Ch. 6.6 - Revising the Predator-Prey Equations Consider a...Ch. 6 - Determining a Solution Determine whether the...Ch. 6 - Determining a Solution Determine whether the...Ch. 6 - Finding a General Solution In Exercises 3-8, use...Ch. 6 - Finding a General SolutionIn Exercises 38, use...Ch. 6 - Finding a General Solution In Exercises 3-8, use...Ch. 6 - Finding a General Solution In Exercises 38, use...Ch. 6 - Finding a General Solution In Exercises 3-8, use...Ch. 6 - Finding a General Solution In Exercises 3-8, use...Ch. 6 - Slope Field In Exercises 9 and 10, a differential...Ch. 6 - Slope Field In Exercises 9 and 10, a differential...Ch. 6 - Slope Field In Exercises 11 and 12, (a) sketch the...Ch. 6 - Slope Field In Exercises 11 and 12, (a) sketch the...Ch. 6 - Euler's Method In Exercises 13 and 14, use Eulers...Ch. 6 - Euler's Method In Exercises 13 and 14, use Eulers...Ch. 6 - Solving a Differential Equation In Exercises 1520,...Ch. 6 - Solving a Differential Equation In Exercises 1520,...Ch. 6 - Solving a Differential Equation In Exercises 1520,...Ch. 6 - Solving a Differential Equation In Exercises 1520,...Ch. 6 - Solving a Differential Equation In Exercises...Ch. 6 - Solving a Differential Equation In Exercises...Ch. 6 - Writing and Solving a Differential Equation In...Ch. 6 - Writing and Solving a Differential Equation In...Ch. 6 - Finding an Exponential Function In Exercises...Ch. 6 - Finding an Exponential Function In Exercises...Ch. 6 - Finding an Exponential Function In Exercises...Ch. 6 - Finding an Exponential Function In Exercises...Ch. 6 - Air Pressure Under ideal conditions, air pressure...Ch. 6 - Radioactive Decay Radioactive radium has a...Ch. 6 - Population A population grows exponentially at a...Ch. 6 - Compound InterestFine the balance in an account...Ch. 6 - Sales The sales S (in thousands of units) of a new...Ch. 6 - Sales The sales 5 (in thousands of units) of a new...Ch. 6 - Finding a General Solution Using Separation of...Ch. 6 - Finding a General Solution Using Separation of...Ch. 6 - Finding a General Solution Using Separation of...Ch. 6 - Finding a General Solution Using Separation of...Ch. 6 - Finding a Particular Solution Using Separation of...Ch. 6 - Finding a Particular Solution Using Separation of...Ch. 6 - Finding a Particular Solution Using Separation of...Ch. 6 - Finding a Particular Solution Using Separation of...Ch. 6 - Slope Field In Exercises 43 and 44, sketch a few...Ch. 6 - Slope Field In Exercises 43 and 44, sketch a few...Ch. 6 - Using a Logistic Equation In Exercises 49 and 50,...Ch. 6 - Using a Logistic Equation In Exercises 49 and 50,...Ch. 6 - Solving a Logistic Differential Equation In...Ch. 6 - Solving a Logistic Differential Equation In...Ch. 6 - Environment A conservation department releases...Ch. 6 - Environment Write a logistic differential equation...Ch. 6 - Sales Growth The rate of change in sales 5 (in...Ch. 6 - Sales Growth Use the result of Exercise 55 to...Ch. 6 - Learning Theory In Exercises 57 and 58, assume...Ch. 6 - Learning Theory In Exercises 57 and 58, assume...Ch. 6 - Solving a First-Order Linear Differential Equation...Ch. 6 - Solving a First-Order Linear Differential Equation...Ch. 6 - Solving a First-Order Linear Differential Equation...Ch. 6 - Solving a First-Order Linear Differential...Ch. 6 - Solving a First-Order Linear Differential...Ch. 6 - Solving a First-Order Linear Differential...Ch. 6 - Solving a First-Order Linear Differential Equation...Ch. 6 - Solving a First-Order Linear Differential Equation...Ch. 6 - Slope Field In Exercises 67-70, (a) sketch an...Ch. 6 - Slope Field In Exercises 67-70, (a) sketch an...Ch. 6 - Slope Field In Exercises 67-70, (a) sketch an...Ch. 6 - Slope Field In Exercises 67-70, (a) sketch an...Ch. 6 - Finding a Particular Solution In Exercises 71-74,...Ch. 6 - Finding a Particular Solution In Exercises 71-74,...Ch. 6 - Writing In Exercises 6769. write an example of the...Ch. 6 - Writing In Exercise 6769. write an example of the...Ch. 6 - Writing In Exercises 6769, write an example of the...Ch. 6 - Investment Let A(t) be the amount in a fund...Ch. 6 - Investment A retired couple plans to withdraw P...Ch. 6 - Investment Use the result of Exercise 70 to find...Ch. 6 - Analyzing Predator-Prey Equations In Exercises 79...Ch. 6 - Analyzing Predator-Prey Equations In Exercises 79...Ch. 6 - Analyzing Competing-Species Equations In Exercises...Ch. 6 - Analyzing Competing-Species Equations In Exercises...Ch. 6 - Doomsday Equation The differential equation where...Ch. 6 - Sales Let S represent sales of a new product (in...Ch. 6 - Error Using the Product Rule Although it is true...Ch. 6 - Torricellis Law Torricellis Law states that water...Ch. 6 - Torricelli's Law The cylindrical water tank shown...Ch. 6 - Torricelli's Law A tank similar to the one in...Ch. 6 - Rewriting the Logistic Equation Show that the...Ch. 6 - Biomass Biomass is a measure of the amount of...Ch. 6 - Finding a Function Consider a function f such that...Ch. 6 - If the tracer is injected instantaneously at time...Ch. 6 - Use the solution of the differential equation in...Ch. 6 - In Exercises 11 and 12, it was assumed that there...

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Prove the identity. 18. 1+tanhx1tanhx=e2x

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