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

Homework 12 Team Problem With the function P(t)=1/(1+e^-t), we can change the height of the horizontal asymptote on the right by changing the value of the numerator 1 to any value. We will use the value of 10, which we wil denote by K. To change the y-intercept to a number between 0 and K, we can change any part of the function to make it have a different y-value when the x-value is 0. We will multiply e^-t by 3. When we plug in 0 for t into the new function P(t)=10/(1+3e^-t), we get 2.5 for the y-intercept value, which is between 0 and K. BONUS After testing different things, we discovered that adding a coefficient to -t will change the slope of the curved part. It must still remain negative. The larger the value of the coefficient, the closer to vertical the slope will be. Compare the graph of the original function to the graph of P(t)=1/(1+e^-5t). You can observe from the graphs below that the windows they are graphed in are different, showing that the slope has actually become more vertical while maintaining the same y-intercept and rest of the graph. The modified function's curved slope is graphed over a smaller range of x-values than the unmodified function's curved slope, showing the change in slope. Changing the coefficient to larger values such as 1/(1+e^-10t) or even larger would make the slope even more vertical. P(t)=1/(1+e^-t) P(t)=1/(1+e^-5t) One application we could use the graph to model is the population of ants in an anthill. Equation for the population of ants:

P(t)=10/(1+3e^-t)
The population is in thousands, The time is in weeks

First off, we must know what the initial population is at time=0 (T0)
P(0)=10/(1+3e^0)
P(0)= 2.5 (in thousands)

With this equation, we can predict what the population will be in 20 weeks. P(20)=10/(1+3e^-20) P(20)=9,999

Assumptions

It is important to note that this model functions because of the assumption that the percentage change in population per week is constant. It is also important to remember that this anthill is in an isolated location where there is no outside interference with the population (humans etc.)

Explaining the Equation

The 10 in the numerator is the horizontal asymptote. Because this equation is modeling population, the horizontal asymptote, which is 10, shows the carrying capacity for this anthill. The y-intercept for this equation is 2.5, which is found when t is zero. The y-intercept represents the initial population.

All the data used to make this equation is imagined and not taken from anywhere.