Spread of Infection The number of people P(t) (in hundreds)...

Spread of Infection The number of people $P(t)$ (in hundreds) infected $t$ days after an epidemic begins is approximated by $P(t)=\frac{10 \ln (0.19 t+1)}{0.19 t+1}$ When will the number of people infected start to decline?

Transcript

00:01 In this portion, we are given a function p of t is equal to 10 times natural law of 0 .19 plus 1 over 0 .190 plus 1.

00:16 This represents the number of people infected t days after the spread of the pandemic.

00:24 We need to obtain the number of days after which the number of days, after which the number of days, after which the patient infected will start getting decreased.

00:36 Now they will start decreasing after the function p of t, that is number of people infected, has reached a maximum value.

00:47 After that, it will start decreasing.

00:50 Now let us first obtain the number of days after which the number of patients infected would be maximum.

00:59 For that, we'll need to differentiate the given function.

01:02 Refrantiate the given function with this pectivity, that is d over dt of 3 of t is equal to d over dt of 10 times natural of 0 .190 plus 1 over 0 .19 plus 1.

01:22 Solving this we get p dash of t is equal to 10 times of d over dt of natural love of 0 .190 plus 1.

01:34 Over 0 .190 plus 1.

01:39 Further, using the quotient rule, the derivative is given by denominator times the derivative of numerator, the derivative log x is 1 over x, therefore 1 over 0 .291, and then the derivative of 0 .190 is 0 .19, minus numerator derivative of denominator over denominator square further.

02:24 So all in this we get, this will cancel out with this and we are left with 10 times of 0 .19 times 1 minus natural log of 0 .19 plus 1 plus 1 over 0 .19 plus 1 over 0 .19 .000.

02:43 20 plus 1 whole square...

Question

Spread of Infection The number of people $P(t)$ (in hundreds) infected $t$ days after an epidemic begins is approximated by $P(t)=\frac{10 \ln (0.19 t+1)}{0.19 t+1}$ When will the number of people infected start to decline?

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