Course:MATH110/Archive/2010-2011/003/Teams/Neuchatel/Homework 12

Team Homework 12

Starting with the function $P(t)={\frac {1}{1+e^{-t}}}$ Your goal is to modify the function so that we can use it to model a real-life problem. We want to be able to control the following things: Change the height of the horizontal asymptote on the right, we'll denote it by $K$. Change the $y$-intercept to any number between $0$ and $K$ BONUS (just the point below, not what comes after) Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical. (If you graph it, it should be quite clear). Once you've played with the function enough, try to find an application of the graph to model something. It can be anything which starts at a value and then goes to another one (think for a population, it goes from 0 to it's carrying capacity). Explain what you are modelling and how you decide to attribute a numerical value to each of the 2 or 3 parameters that you researched just above. Then use the model to make a prediction. For example, if your model is suppose to describe a population for which you have its initial population and carrying capacity (potentially its rate of increase if you solved the bonus part), then use that data to make a prediction for the population in 20 years, or use the model to predict when will the population reach 95% of its carrying capacity). When doing this last part, explain well where you're taking your data from (real data or imagined data), what it is that you're modelling and how you are doing the math to answer a predictive question.

To change the y-intercept between any number 0 and k and to change the height of the horizontal asymptote on the right we changed the value of x from 1 to 0.1 where:

$P(t)={\frac {1}{x+e^{-t}}}$

The smaller the value of x the larger the value of the y intercept and the horizontal asymptote on the right so we figured that having x= 0.1 would give us a good starting population for our bacteria growth.

From looking at the graph of the original function and our new function you can see the new value of the horizontal asymptote on the right k = 10 and the y intercept = 1

So so far we have the function: $P(t)={\frac {1}{0.1+e^{-t}}}$

To change the slope of the curve so that the slope can go from close to zero to almost vertical we must simply replace the value "e" to a larger value such as "50". This will distinctively change the slope of the curve to a near straight line.

$P(t)={\frac {1}{0.1+50^{-t}}}$

Finally, now that we have come up with a model to describe the population of bacteria with a fixed amount of food available as a function of time, we simply substitute a time value in which we are interested into the equation.At this juncture, we are interested to know what the bacteria population after ${\frac {1}{2}}$ year will be. Hence,

$P({182.5})={\frac {1}{0.1+50^{-182.5}}}=10$

Therefore, the population of bacteria, according to the modified model, is approximately 10 after 1/2 year.