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=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?

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%