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

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^{-(t)/(12)}\\\\ \text{for}\ 0\leq t \leq 48`

`\Rightarrow P'(t)=(d)/(dt)[7te^{-(t)/(12)}]=7[t.(d)/(dt)(e^{-(t)/(12)})+(d)/(dt)(t).e^{-(t)/(12)}]\\\\\Rightarrow P'(t)=7[t.(-(1)/(12)* e^{-(t)/(12)})+1.e^{-(t)/(12)}]\\\\\Rightarrow P'(t)=7[e^{-(t)/(12)}-(t)/(12)e^{-(t)/(12)}}]\\\\\Rightarrow P'(t)=7e^{-(t)/(12)}[1-(t)/(12)}]`

Finding the critical values,

`\Rightarrow P'(t)=0`

`\Rightarrow 7e^{-(t)/(12)}[1-(t)/(12)}]=0`

`\Rightarrow [1-(t)/(12)}]=0`

`\Rightarrow (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^{-(12)/(12)}=7(12)e^(-1)=(84)/(e)=30.90`

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