Mathematics 150AL

Assignment 3

Due: March 11, 2008

Population Growth Exponential Growth: The total change of a population of organisms such as bacteria or yeast between the time $t$ and $\ t+\Delta t$ is given by: MATH In the simplest case a fixed ratio $b$ of the existing organisms will give birth and another fixed ratio $d$ of them will die. $b$ is called the birthrate, $d$ is called the death rate. It follows that$\vspace{1pt}$

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where $\ \alpha =b$-$d$ is called the rate of growth. The factor $\Delta t$ signifies that the total amount of change will be proportional to the length of the interval $\Delta t$. If we divide the above equation by $\Delta t$ and take the limit as MATH0 on both sides we get:

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This is a linear differential equation. We can solve this equation using Scientific Notebook. Type

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and choose Solve ODE+Exact in the Compute menu. In the dialogue box we choose $t$ as independent variable. The computer will give the following answer:

Exact solution is: $C_{1}e^{t\alpha }$

We see that this answer depends on a constant $C_{1}$. The population of the organisms at time $t$ will of course depend on the population of the organisms at

the start. This constant represents the original population. Example: A biologist starts a bacterial culture initially containing 1 million bacteria. The population grows by 20% per hour. In this case the constant $C_{1}=1,$ if measure the population in millions. $\alpha =0.2$. The population is given

$P(t)=e^{0.2t}$

Exercise 1

Often we do not know the growth rate, but we can find it from measured data. In the following exercise you should do this.

A biologist starts a bacterial culture initially containing 1.3 million bacteria. After 45 minutes the culture contains 1.4 million bacteria.

1 Find $\alpha $ for this problem from the given data.

2. Sketch the graph of the population versus time. Pick an appropriate scale.

3. How long will it take the population to double? This is called the doubling time.

4. How long will it take the population to quadruple?

Exercise 2

Newton's Law of Cooling states that the temperature $T$ of a body in an environment of temperature $T_{e}$ is given by

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where $t$ is the time passed since the body introduced in this environment. And $T_{0}$ is the temperature of the body when it was introduced into this environment. The constant $\alpha $ depends on the shape of the body its material (a hot piece of metal cools down faster than a clay brick of the same temperature), and the environment (water is a better coolant than air, for example). This constant has to be determined experimentally in each case.

Exercise 2: (A murder mystery) On a warm and sunny morning the dead body of a formerly famous actress is found floating in the pool of her estate in the Hollywood Hills.

Pretend that you are a forensic scientist working for the coroner's office. It is your task to determine the time of death.

You arrive at 8 a.m. and measure the body temperature at 25 C. You also observe that the heated pool is kept at precisely 20 C. You assume that the victim's temperature was normal before death. (Normal body temperature is

36.5 C) Thirty minutes after your arrival, you remeasure the temperature of the corpse: it is 24.9 C. Write an initial report in which you indicate the time of the murder.

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