(A) Explanation Given: The epidemic had spread throughout a community of 50,000 people. The rate at which the people get infected is proportional to the product of the number of people who have been infected and the people who have not been infected. The initial number of individuals affected was 100. After 10 days, the number of infected people is 500. Theorem used: Separation of Variables: The solution of the differential equation, f ( y ) y ′ = g ( x ) is given implicitly by the equation, ∫ f ( y ) d y = ∫ g ( x ) d x . Formula used: Logistic growth model: If the rate of change with respect to time t of a quantity y is proportional to both the amount present and the difference between y and a limiting value M , then y satisfies the differential equation, d y d t = k y ( M − y ) , k > 0 , t > 0 Calculation: Let y represents the number of people who have been infected. The total number of people in the community is 50,000. Thus, the number of people who have not been infected will be 50 , 000 − y . The rate is proportional to the product of infected people and uninfected people, which is d y d t = k y ( 50 , 000 − y ) , k > 0 , t > 0 . Thus, the problem follows a logistic growth model. Rewrite the differential equation d y d t = k y ( 50 , 000 − y ) as follows. d y y ( 50 , 000 − y ) = k d t 1 y ( 50 , 000 − y ) d y = k d t The given differential equation d y d t = k y ( 50 , 000 − y ) is in the form of f ( y ) y ′ = g ( x ) where, f ( y ) = 1 y ( 50 , 000 − y ) and g ( t ) = k . By the above mentioned theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ ( 1 y ( 50 , 000 − y ) ) d y = ∫ k d t (1) Rewrite the integral ∫ ( 1 y ( 50 , 000 − y ) ) d y as, ∫ ( 1 y ( 50 , 000 − y ) ) d y = ∫ ( 50 , 000 + y − y 50 , 000 y ( 50 , 000 − y ) ) d y = 1 50 , 000 ∫ ( 1 y + 1 50 , 000 − y ) d y Obtain the value of each integral as follows. 1 50 , 000 ∫ ( 1 y + 1 50 , 000 − y ) d y = 1 50 , 000 [ ln y − ln ( 50 , 000 − y ) ] Substitute the value of the integral ∫ ( 1 y ( 50 , 000 − y ) ) d y = 1 50 , 000 [ ln y − ln ( 50 , 000 − y ) ] in equation (1) and simplify as follows. ∫ ( 1 y ( 50 , 000 − y ) ) d y = ∫ k d t 1 50 , 000 [ ln y − ln ( 50 , 000 − y ) ] = k t + C 1 ln y − ln ( 50 , 000 − y ) = 50 , 000 ( k t + C 1 ) ln ( y 50 , 000 − y ) = 50 , 000 ( k t + C 1 ) Simplify further as, e ln ( y 50 , 000 − y ) = e 50 , 000 ( k t + C 1 ) y (B) To determine To find: The time taken for the disease to spread around half the community.

14th Edition

ISBN: 9780134856667

Author: Stocker

Publisher: VST

See similar textbooks

Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Videos

Question

Chapter 9.2, Problem 77E

(A)

To determine

To find: The number of people who have been infected after 20 days.

(B)

To determine

To find: The time taken for the disease to spread around half the community.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.2, Problem 76E

Chapter 9.2, Problem 78E

Students have asked these similar questions

Business Discuss

Find the derivative of the function. 5 1 6 p(x) = -24x 5 +15x

∞ 2n (4n)! Let R be the radius of convergence of the series -x2n. Then the value of (3" (2n)!)² n=1 sin(2R+4/R) is -0.892 0.075 0.732 -0.812 -0.519 -0.107 -0.564 0.588

Chapter 9 Solutions

EBK CALCULUS FOR BUSINESS, ECONOMICS, L

Ch. 9.1 - Show that y=Cx+1 is the general solution of the...Ch. 9.1 - Prob. 2MP Ch. 9.1 - Prob. 3MP Ch. 9.1 - Prob. 4MP Ch. 9.1 - Prob. 5MP Ch. 9.1 - Prob. 1ED Ch. 9.1 - Prob. 2ED Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The number of people who have been infected after 20 days.

Solution Summary: Theorem used: Logistic growth model: if the rate of change with respect to time t is proportional to both the amount present and the difference between y and a limiting value