There are 150000 people in the population who hear the rumor eventually. 9 percent of them heard it on the first day. Find k , assuming t is measured in days.

Question

The number, N , of people who have heard a rumor spread by mass media at time, t , is given by

N(t) = a(1 - e ^-kt)

There are 150000 people in the population who hear the rumor eventually. 9 percent of them heard it on the first day. Find k , assuming t is measured in days.

k =

Answer 1

When t gets very large (that is "eventually"), then e^-kt approaches zero, meaning that "a" must be 150,000.

So N(t) = 150,000(1 - e^-kt)

Since 150,000•0.09 = 13,500, then the point (1, 13,500) is a point on the curve meaning:

N(1) = 13,500 = 150,000(1 - e^-k(1))

13,500/150,000 = 1 - e^-k

0.09 = 1 - e^-k

e^-k = 1 - 0.09

e^-k = 0.91

ln(e^-k) = ln(0.91)

-k = ln(0.91)

k = -ln(0.91)

A decimal approximation is that k ≈ -0.09431