Question

A disease 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?

Answer 1

t = 13 days

p(13) = 33.47%

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%