Question

A flu has hit a city and the percentage of a population with the flu, t days after the disease arrives is approximated by f(t)=10te^(-t/8) for 0<=t>=40.
1)after how many days is the percentage of the population with the flu a maximum?
2)what is the maximum percent of the population at this time?

Answer 1

1) To calculate maximum of f(t) function we first need to find derivative of it:
f(t)' = 10(e^(-t/8) + t*e^(-t/8)*(-1/8)) = 10(e^(-t/8) -t/8*e^(-t/8)) = 10e^(-t/8)(1-t/8)

the condition is:
f(t)' = 0 that means:
0 = 10e^(-t/8)(1-t/8) 
10t/8*e*(-t/8) = 10*e^(-t/8)
t/8 = 1
t = 8

The answer is 8 days.

2) that percent we will get simply by expressing t=8 in our equation.
f(8) = 10*8*e^(-1) = 80/e = 29.43%