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Len [333]
3 years ago
13

A culture of the bacterium Salmonella enteritidis initially contains 50 cells. When introduced into a nutrient broth, the cultur

e grows at a rate proportional to its size. After 1.5 hours, the population has increased to 775. (a) Find an expression for the number of bacteria after t hours. (Round your numeric values to four decimal places.)
Mathematics
1 answer:
photoshop1234 [79]3 years ago
6 0

Answer:

The expression for the number of bacteria after t hours is P(t) = 50e^{0.2922t}

Step-by-step explanation:

When introduced into a nutrient broth, the culture grows at a rate proportional to its size.

This means that the size of the population, after t hours, is modeled by the following differential equation:

\frac{dP}{dt} = rP

In which R is the growth rate.

The solution of this differential equation is:

P(t) = P(0)e^{rt}

In which P(0) is the initial population.

A culture of the bacterium Salmonella enteritidis initially contains 50 cells.

This means that P(0) = 50, and so:

P(t) = P(0)e^{rt}

P(t) = 50e^{rt}

After 1.5 hours, the population has increased to 775.

This means that P(1.5) = 775. We use this to find r. So

P(t) = P(0)e^{rt}

500e^{1.5r} = 775

e^{1.5r} = \frac{775}{500}

\ln{e^{1.5r}} = \ln{\frac{775}{500}}

1.5r = \ln{\frac{775}{500}}

r = \frac{\ln{\frac{775}{500}}}{1.5}

r = 0.2922

The expression is:

P(t) = 50e^{rt}

P(t) = 50e^{0.2922t}

The expression for the number of bacteria after t hours is P(t) = 50e^{0.2922t}

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