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?

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%