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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. After 2 hours, there are 200 bacteria, and after 4 hours, there are 300 bacteria.

(a) Find an exponential model for the population P after t hours.

(b) How many bacteria will there be after 7 hours?

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

(a)

To determine

To calculate: The exponential model for the population P after t hours the rate of increasing a certain type of bacteria is proportional to the number present and the number of bacteria after 2 hours and after 4 hours are 200 and 300.

Explanation

Given Information:

The rate of increasing a certain type of bacteria is proportional to the number present and the number of bacteria after 2 hours and after 4 hours are 200 and 300.

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 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 exponential decay.

If the expression is in the form of ex=b, then the property of natural logarithms is given by.

lnb=x

Calculation:

Consider the equation N=Cekt.

Here, the population P at t=2 is 200

Substitute in the equation N=Cekt

200=Cek2200=Ce2k

Here, the population P at t=4 is 300

Substitute in the equation N=Cekt

300=Cek4300=Ce4k

Solve the two equation for the value of C

(b)

To determine

To calculate: The number of bacteria after 7 hours the rate of increasing a certain type of bacteria is proportional to the number present and the number of bacteria after 2 hours and after 4 hours are 200 and 300.

(b)

To determine

To calculate: The time required to double the population in size if the rate of increasing a certain type of bacteria is proportional to the number present and the number of bacteria after 2 hours and after 4 hours are 200 and 300.

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