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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Population Growth The number of a certain type of bacteria increases continuously at a rate proportional to the number present. There are 150 bacteria at a given time and 450 bacteria 5 hours later.

(a) How many bacteria will there be 10 hours after the initial time?

(b) How long will it take for the population to double?

(c) Does the answer to part (b) depend on the starting time? Explain your reasoning.

(a)

To determine

To calculate: The number of bacteria after 10 hours if there are 150 bacteria at a given time and 450 bacteria after 5 hours.

Explanation

Given Information:

The provided information is that there are 150 bacteria at a given time and 450 bacteria 5 hours later.

Formula used:

Exponential growth and decay:

If the rate of change of a positive quantity y with respect to time is proportional to the amount of quantity present at any time t, that is dydt=ky, then y is given by the equation, y=Cekt, where C is the value of the quantity at a time t=0 and k is the constant of proportionality.

If k>0, then there is exponential growth and when k<0, then there is an exponential decay.

Calculation:

Consider the provided information that there are 150 bacteria at a given time and 450 bacteria 5 hours later.

Let t=0 when there are 150 bacteria.

Hence, an initial number of bacteria at t=0 is 150.

So, C=150.

Substitute C=150 in the equation y=Cekt,

y=150ekt

Now, 5 hours later, there are 450 bacteria.

So, when t=5 y=450.

Substitute t=5 and y=450 in the equation y=150ekt

(b)

To determine

To calculate: The time to double the population of bacteria if there are 150 bacteria at a given time and 450 bacteria 5 hours later.

(c)

To determine

If the time to double the population of bacteria as calculated in part (b) is dependent on the starting time provided that there are 150 bacteria at a given time and 450 bacteria 5 hours later.

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