What is the maximum percent of the population infected?

Question

The percentage of the population infected t days after the disease arrives is approximated by p(t)= 8t e ^ -t/11 for 0 < t <. After how many days is the percentage of infected people a maximum? What is the maximum percent of the population infected?

Answer 1

p(t)= 8t e^-t/11

 

Assuming this is a Calculus question, take the derivative of p(t) wrt t, set it to zero, and solve for t.  Use the Product and Chain rules to take the derivative:

 

dp(t)/dt = 8e^-t/11 - (8t/11)e^-t/11 = (1 - t/11)*8e^-t/11

 

0 = (1 - t/11)*8e^-t/11

 

which occurs when t = 11 days, which is when the infection affects the maximum percentage of the population.  The max percentage is:

 

p(11) = 8*11 e^-11/11 = 88/e ≈ 32.4%