Adisease has hit a city. the percentage of the population infected t days after the disease arrives is approximated by p(t)equals7 t e superscript negative t divided by 13 for 0less than or equalstless than or equals39. after how many days is the percentage of infected people a maximum? what is the maximum percent of the population infected?

t = 13 days

p(13) = 33.47%

Step-by-step explanation:

p(t) is the percentage of the population infected:

p(t) = 7*t*e∧(-t / 13)

where    0 ≤ t ≤ 39 days

we can apply p'(t) = 0  to get number of days where the percentage of infected people is maximum:

p'(t) = (7*t*e∧(-t / 13))' = 7*(t*e∧(-t / 13))' = 7*((t)'*e∧(-t / 13)+t*(e∧(-t / 13)') = 0

⇒  7*(1*e∧(-t / 13)+t*e∧(-t / 13)*(-1 / 13)) = 7*e∧(-t / 13)*(1 - (t / 13)) = 0

∴  1 - (t / 13) = 0    ⇒    t = 13 days

then we get the maximum percent of the population infected as follows

p(13) = 7*13*e∧(-13 / 13)

⇒  p(13) = 33.47%