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 12 for 0less than or equalstless than or equals48. After how many days is the percentage of infected people a​ maximum? What is the maximum percent of the population​ infected? The percentage of infected people reaches a maximum after how many days.

Answer 1

Answer-

After 12 days the percentage of infected people will maximum and the maximum value will be 30.90%

Solution-

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

$P(t)=7te^{-\frac{t}{12}}\\\\ \text{for}\ 0\leq t \leq 48$

$\Rightarrow P'(t)=\frac{d}{dt}[7te^{-\frac{t}{12}}]=7[t.\frac{d}{dt}(e^{-\frac{t}{12}})+\frac{d}{dt}(t).e^{-\frac{t}{12}}]\\\\\Rightarrow P'(t)=7[t.(-\frac{1}{12}\times e^{-\frac{t}{12}})+1.e^{-\frac{t}{12}}]\\\\\Rightarrow P'(t)=7[e^{-\frac{t}{12}}-\frac{t}{12}e^{-\frac{t}{12}}}]\\\\\Rightarrow P'(t)=7e^{-\frac{t}{12}}[1-\frac{t}{12}}]$

Finding the critical values,

$\Rightarrow P'(t)=0$

$\Rightarrow 7e^{-\frac{t}{12}}[1-\frac{t}{12}}]=0$

$\Rightarrow [1-\frac{t}{12}}]=0$

$\Rightarrow \frac{t}{12}}=1$

$\Rightarrow t=12$

Therefore, after 12 days the percentage of infected people will be​ maximum

And maximum value will be,

$P(12)=7(12)e^{-\frac{12}{12}}=7(12)e^{-1}=\dfrac{84}{e}=30.90$

Therefore, after 12 days the percentage of infected people will maximum and the maximum value will be 30.90%