   Chapter 12.5, Problem 44E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Bacterial growth Suppose that for a certain population of bacteria, growth occurs according to d y d t = k y ( t  in hours,  k  constant ) If the doubling rate depends on temperature, find how long it takes for the number of bacteria to reach 50 times the original number at each given temperature in parts (a) and (b).(a) At 90°F , the number doubles after 30 minutes ( 1 2  hour ) .(b) At 40°F , the number doubles after 3 hours.

(a)

To determine

To calculate: The time required to reach number of bacteria 50 times the original number at 90° F.

Explanation

Given Information:

The certain population of bacteria, growth occurs according to

dydt=ky

If the doubling rate depends on the temperature, 10,000 organism and the number triples after 2 hours.

The bacteria will be double after 30 minute.

Formula used:

The differential equation can be equivalently expressed in the form

g(y)dy=f(x)dx

Then the equation is separable.

The solution of a separable differential is obtained by integrating both sides of the equation after the variable have been separated.

Calculation:

Consider the certain population of bacteria, growth occurs according to

dydt=ky

If the doubling rate depends on the temperature, 10,000 organism and the number triples after 2 hours.

Firstly, separate the above differential equation as:

dyy=kdt

Integrate the provided function as:

dyy=kdtln|y|=kt+Cy=Cekt

At the given condition t=0 put in the above equation as:

(b)

To determine

To calculate: The time required to reach number of bacteria 50 times the original number at 40° F.

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