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 8 for 0less than or equalstless than or equals32. After how many days is the percentage of infected people a​ maximum? What is the maximum percent of the population​ infected?

Answer 1

Answer:

(1) The percentage of infected people will be a​ maximum after 8 days.

(2) The maximum percent of the population​ infected is 20.60%.

Step-by-step explanation:

The percentage of the population infected t days after the disease arrives is approximated by:

$p(t)=7te^{-t/8};\ 0\leq t\leq 32$

(1)

The percentage of infected people will be a​ maximum when p' (t) = 0.

Compute the value of p' (t) and equate it to 0 as follows:

$p '(t) = 7(1)e^{-t/8} +7te^{-t/8}(-1/8)$

    $0=e^{-t/8}(7 -\frac{7t}{8})$

    $0=7-\frac{7t}{8}\\0=56-7t\\7t=56\\t=8$

Thus, the percentage of infected people will be a​ maximum after 8 days.

(2)

Compute the value of p (8) as follows:

$p(8)=7\times8\times e^{-8/8}\\=7\times 8\times 0.36788\\=20.60128\\\approx 20.60\%$

Thus, the maximum percent of the population​ infected is 20.60%.