#5. If a rancher has one cow with a contagious disease in a herd of 1000, then the time in days “t” for “n” of the cows to become infected is modelled by:
t= -5ln((1000-n)/999n
where “ln()” stands for natural logarithm function.
a) Find the number of days that it takes for the disease to spread to 100, 200, 998, and 999 cows.
b) Graph the number of infected cows in the x-axis versus the time in days it takes for infection in the y-axis.
This model, called a logistic growth model, describes how a disease can spread very rapidly at first and then very slowly as nearly all of the population has become infected.
100, 23.55
200, 27.6
998, 65.6
999, 69,1 and you can find points in between to make your graph, It will be an upward graph to the left
Sorry to the right
To find the number of days it takes for the disease to spread to a specific number of cows, we can substitute the given values of "n" into the equation and calculate the value of "t". Let's calculate the number of days it takes for the disease to spread to 100, 200, 998, and 999 cows:
a) For n=100:
t = -5ln((1000-100)/999*100)
t = -5ln(900/99900)
t ≈ 5.83 days
b) For n=200:
t = -5ln((1000-200)/999*200)
t = -5ln(800/199800)
t ≈ 6.52 days
c) For n=998:
t = -5ln((1000-998)/999*998)
t = -5ln(2/998002)
t ≈ 5.00 days
d) For n=999:
t = -5ln((1000-999)/999*999)
t = -5ln(1/998001)
t ≈ 0.00 days
To graph the number of infected cows on the x-axis versus the time in days on the y-axis, we can create a table with different values of "n" and calculate the corresponding values of "t". Then, we can plot the points on the graph.
Here is the graph:
Note that the x-axis represents the number of infected cows (n) and the y-axis represents the time in days (t). As mentioned in the question, the disease spreads rapidly at first and then slows down as most of the population becomes infected.