Calculus Question

Question

Assume 8 people originally have the virus and that in early stages the number of people infected is increasing approximately exponentially, with a continuous growth rate of 1.06. Assume further that in the long run, they estimate approximately 6000 people will be infected. 

(You may wish to sketch a graph of your logistic function P=L/1+Ce^−kt to help you understand this situation and answer the questions below.) 

(a) What should we use for the parameters k and L?  

k= 

L= 

 

(b) Use the fact that when t=0, we have P=8, to find C. 

C= 

(c) What is the value of P when the rate at which people are becoming infected peaks? 

P=

 

When does this occur? 

t= 

Answer 1

L = 6000

k = 1.06

6000/1+Ce^-1.06(0) = 6000/1+C = 8

8+8C = 6000

8C = 5992

C = 749

The disease is a maximum when P = 6000/2 = 3000; this result is known for the logistic function, so you don't have to determine when the second derivative equals 0.

6000/1+749e^-1.06t = 3000

(1+749e^-1.06t)3000 = 6000

(1+749e^-1.06t) = 2

749e^-1.06t = 1

e^-1.06t = 1/749 or

e^1.06t = 749

t = ln(749)/ln(1.06) = 113.5895