Question

The number of people who have heard a rumor is modeled by the logistic function f(x)=2601+23e−0.06x

, where x is the number of days since the rumor began.

Approximately how many days have passed when the rate at which the rumor spreads begins to decrease?

Answer 1

C

Explanation:

The number of people who have heard a rumor is modeled by the logistic function:

`\displaystyle f(x)=(260)/(1+23e^(-0.06x))`

And we want how many days has passed since the rate at which the rumor spread began to decrease.

First, the numerator tells us that the total number of people who will have heard the rumor by the end is 260.

Recall that the point of maximum growth is always half of the maximum capacity.

In this case, that point will be 260/2 = 130.

For a logistic function, the rate of all values to the left of the halfway point is increasing, (i.e. concave up), while the rate of all values to the right is decreasing (i.e. concave down). Therefore, we can solve for x when f(x) = 130:

`\displaystyle 130=(260)/(1+23e^(-0.06x))`

Cross-multiply:

`130(1+23e^(-0.06x)})=260`

Distribute:

`130+2990e^(-0.06x)=260`

Solving for x:

`2990e^(-0.06x)=130`

Therefore:

`\displaystyle e^(-0.06x)=(130)/(2990)=(1)/(23)`

Taking the natural log of both sides:

`\displaystyle -0.06x=\ln\Big((1)/(23)\Big)`

Therefore:

`\displaystyle x=(\ln(1/23))/(-0.06)\approx52.3`

After about 52.3 days, the rate at which the rumor started spreading will begin to decrease.