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3 January, 03:01

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?

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  1. 3 January, 03:27
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    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%
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