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Bacteria Growth At time t = 0 , a bacterial culture weighs 1 gram. Two hours later, the culture weighs 4 grams. The maximum weight of the culture is 20 grams. (a) Write a logistic equation that models the weight of the bacterial culture. (b) Find the culture’s weight after 5 hours. (c) When will the culture’s weight reach 18 grams? (d) Write a logistic differential equation that models the growth rate of the culture’s weight. Then repeat part (b) using Euler’s Method with a step size of h = 1 . Compare the approximation with the exact answer. (e) After how many hours is the culture’s weight increasing most rapidly? Explain.

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Calculus: Early Transcendental Fun...

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

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285774770
Chapter 6.4, Problem 32E
Textbook Problem
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Bacteria Growth At time t = 0 , a bacterial culture weighs 1 gram. Two hours later, the culture weighs 4 grams. The maximum weight of the culture is 20 grams.

(a) Write a logistic equation that models the weight of the bacterial culture.

(b) Find the culture’s weight after 5 hours.

(c) When will the culture’s weight reach 18 grams?

(d) Write a logistic differential equation that models the growth rate of the culture’s weight. Then repeat part (b) using Euler’s Method with a step size of h = 1 . Compare the approximation with the exact answer.

(e) After how many hours is the culture’s weight increasing most rapidly? Explain.

(a)

To determine

To calculate: The logistic equation that models the weight of the bacteria culture where the pattern is given as follows:

At time t=0, a bacterial culture weighs 1 gram. Two hours later, the culture weighs 4 grams. The maximum weight of the culture is 20 grams.

Explanation of Solution

Given:

At time t=0, a bacterial culture weighs 1 gram. Two hours later, the culture weighs 4 grams. The maximum weight of the culture is 20 grams.

Formula used:

The general logistic equation is y=L1+bekt.

Calculation:

Consider the expression:

y=L1+bekt.…...…... (1)

Given that at t=0, y=1 and L=20. Put all values in equation (1):

y=L1+bekt1=201+bek×01=201+b1+b=20

Further simplify,

b=19

Also, given that at t=2, y=4 and L=20

(b)

To determine

To calculate: The culture weight after 5 years.

(c)

To determine

To calculate: The time when the weight will reach 18 g.

(d)

To determine

To calculate: The logistic differential equation that models growth rate of the culture’s weight. Also, use Euler’s method with step size of h=1 to find culture’s weight after 5 years and compare the approximation with exact value.

(e)

To determine

To calculate: The hours after which the culture’s weight is increasing most rapidly.

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Chapter 6 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)
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The graph shows the amount of...Ch. 6.5 - Standard Form Give the standard form of a...Ch. 6.5 - Matching In Exercises 43-46. match the...Ch. 6.5 - Matching In Exercises 43-46. match the...Ch. 6.5 - Matching In Exercises 43-46. match the...Ch. 6.5 - Matching In Exercises 43-46. match the...Ch. 6.5 - Slope Field In Exercises 4750, (a) use a graphing...Ch. 6.5 - Slope Field In Exercises 47 and 48, (a) use a...Ch. 6.5 - Slope Field In Exercises 4750, (a) use a graphing...Ch. 6.5 - Slope Field In Exercises 4750, (a) use a graphing...Ch. 6.5 - Solving a First-Order Differential Equation In...Ch. 6.5 - Solving a First-Order Linear Differential Equation...Ch. 6.5 - Solving a First-Order Differential Equation In...Ch. 6.5 - Solving a First-Order Differential Equation In...Ch. 6.5 - Solving a First-Order Differential Equation In...Ch. 6.5 - Solving a First-Order Differential Equation In...Ch. 6.5 - Solving a First-Order Differential Equation In...Ch. 6.5 - Solving a First-Order Differential Equation In...Ch. 6.5 - Solving a Bernoulli Differential Equation In...Ch. 6.5 - Solving a Bernoulli Differential Equation In...Ch. 6.5 - Solving a Bernoulli Differential EquationIn...Ch. 6.5 - Solving a Bernoulli Differential EquationIn...Ch. 6.5 - Solving a Bernoulli Differential Equation In...Ch. 6.5 - Solving a Bernoulli Differential Equation In...Ch. 6.5 - Solving a Bernoulli Differential Equation In...Ch. 6.5 - Solving a Bernoulli Differential Equation In...Ch. 6.5 - True or False? 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 - 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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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