Course:MATH110/Archive/2010-2011/003/Teams/Uri/Homework/12 Part3
Homework
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
- 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).
Original function:
Which is shown as:
Change the height of the horizontal asymptote on the right, we'll denote it by K.
P(t)= k /[1+e^-x]
P(t)= 5 /[1+e^-x]
P(t)= 10 /[1+e^-x]
As we can see the horizontal asymptote has been shifted higher.
Which is shown as:
The larger the number for K the horizontal asymptote increases.
Lower Horizontal asymptote: P(t)= 5 /1+e^-x]
High Horizontal asymptote: P(t)= 10 /[1+e^-x]
Change the y-intercept to any number between 0 and K
P(t)= 1 /[(1+ e^(-x+y))]
P(t)= 1 /[(1+(e^(-t+5))]
P(t)= 1/(1+(e^(-t+1))]
Which is shown as:
By changing the value of Y into a larger number it can be observed that the Y-intercept becomes less and less.
This one has a lower Y-Intercept: P(t)= 1 /[(1+(e^(-t+5))]
This one has a higher Y-Interecept: P(t)= 1/(1+(e^(-t+1))]
Applying the Model
An eco-system which is supporting a population of birds can theoretically can hold up to 300 birds, which is its carrying capacity. Initially the population of birds is 50 with a growth rate of .6 each year. Using the model:
We can input the data given to form:
P(t)= C/1+Ie^(-rt)
C = Carrying capacity of the eco-system
I = Initial starting population
R = Rate of growth
P(t)= 300/1+50e^(-.6t)
For the predictions how long does it take for the population of bird to reach 80% of its carrying capacity?
To find this we use the function: P(t)= 300/1+50e^(-.6t)
P= Population
t= Time it takes for the population to grow
300*.80= 540
So we are looking for how long its take for the population to grow to 540 Birds. To find that we plug 540 into P:
540= 300/1+50e^(-.6t)
Then Solve for T
