Biology The population y of a bacterial culture is modeled by the logistic growth function y = 925 / ( 1 + e − 0.3 t ) , where t is the time in days. (a) Use a graphing utility to graph the model. Find the number of days it takes for the population to reach 711. (b) Does the population have a limit as t increases without bound? Explain your answer. (c) How does the limit change when the model is y = 1000 / ( 1 + e − 0.3 t ) ? Explain your answer. Draw some conclusions about this type of model.

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(a) To determine To graph: The model by using technology which represents the population y of a bacterial culture increases with the time t (in days) is given by the logistic growth function as, y = 925 ( 1 + e − 0.3 t ) Also determine the number of days required in order that the population of the culture will reach 711 . Explanation Given Information: The model which represents the population y of a bacterial culture increases with the time t (in days) is given by the logistic growth function as, y = 925 ( 1 + e − 0.3 t ) Graph: Consider the model, y = 925 ( 1 + e − 0.3 t ) Graph of the model using the TI- 83 graphing calculator is as follows: Step 1 . Press “ Y = ” key and insert the function as, Y 1 = 925 ÷ ( 1 + e ^ ( - 0.3 X)) Step 2 . Press the “WINDOW” key and set the values as, X min = − 10,Xmax = 30 , Xscl = 1 , Ymin = − 100,Ymax = 1200 , Yscl = 1 , Xres = 1 Step4. Then press the “TRACE” key. A window showing the graph of the pair of functions will appear as, Compute the number of days required in order that the population of the culture will reach 711 by substituting y = 711 in the function as, 711 = 925 ( 1 + e − 0.3 t ) Multiply both sides by ( 1 + e − 0.3 t ) as, 711 ( 1 + e − 0.3 t ) = 925 ( 1 + e − 0.3 t ) ⋅ ( 1 + e − 0.3 t ) 711 ( 1 + e − 0.3 t ) = 925 Divide both sides by 711 as, 711 ( 1 + e − 0.3 t ) 711 = 925 711 1 + e − 0 (b) To determine Whether the limit of population of bacterial culture have some value or not as t increases without bound and give reason when the population y of a bacterial culture increases with the time t (in days) is given by the logistic growth function as, y = 925 ( 1 + e − 0.3 t ) (c) To determine To calculate: The limit of population of bacterial culture and give interpretation about the type of model when the population y of a bacterial culture increases with the time t (in days) is given by the function as, y = 1000 ( 1 + e − 0.3 t )

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Calculus: An Applied Approach (Min...

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Ron Larson

Publisher: Cengage Learning

ISBN: 9781305860919

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BuyFind

Calculus: An Applied Approach (Min...

10th Edition

Ron Larson

Publisher: Cengage Learning

ISBN: 9781305860919

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Answer 3

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Answer 4

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Answer 5

(a)

To determine

To graph: The model by using technology which represents the population y of a bacterial culture increases with the time t (in days) is given by the logistic growth function as,

y=925(1+e−0.3t)

Also determine the number of days required in order that the population of the culture will reach 711.

Explanation of Solution

Given Information:

The model which represents the population y of a bacterial culture increases with the time t (in days) is given by the logistic growth function as,

y=925(1+e−0.3t)

Graph:

Consider the model, y=925(1+e−0.3t)

Graph of the model using the TI-83 graphing calculator is as follows:

Step 1. Press “Y=” key and insert the function as,

Y1=925÷(1+e^(-0.3X))

Step 2. Press the “WINDOW” key and set the values as,

Xmin=−10,Xmax=30,Xscl=1,Ymin=−100,Ymax=1200,Yscl=1,Xres=1

Step4. Then press the “TRACE” key.

A window showing the graph of the pair of functions will appear as,

Compute the number of days required in order that the population of the culture will reach 711 by substituting y=711 in the function as,

711=925(1+e−0.3t)

Multiply both sides by (1+e−0.3t) as,

711(1+e−0.3t)=925(1+e−0.3t)⋅(1+e−0.3t)711(1+e−0.3t)=925

Divide both sides by 711 as,

711(1+e−0.3t)711=9257111+e−0

Answer 6

(b)

To determine

Whether the limit of population of bacterial culture have some value or not as t increases without bound and give reason when the population y of a bacterial culture increases with the time t (in days) is given by the logistic growth function as,

y=925(1+e−0.3t)

Answer 7

(c)

To determine

To calculate: The limit of population of bacterial culture and give interpretation about the type of model when the population y of a bacterial culture increases with the time t (in days) is given by the function as,

y=1000(1+e−0.3t)

Calculus: An Applied Approach (Min...

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Calculus: An Applied Approach (Min...

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