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 250000 people in the population who hear the rumor eventually. 25 percent of them heard it on the first day. Find a and k , assuming t is measured in days.
To find the values of a and k in the given function N(t) = a(1 - e^(kt)), we can use the information given.
Let's break down the information provided:
1. The number of people who have heard the rumor eventually is 250,000.
This means that when we substitute t = infinity (since eventually all people hear the rumor), N(t) = 250,000.
N(t = infinity) = a(1 - e^(k * infinity)) = 250,000
Note that e^(k * infinity) approaches infinity as t approaches infinity, making the expression 1 - e^(k * infinity) equal to 1 - (infinity), which is 1.
Therefore, we have:
a(1 - 1) = 250,000
This simplifies to:
a * 0 = 250,000
Since any number multiplied by 0 is 0, we conclude that a = 0.
2. 25 percent of the population heard the rumor on the first day.
When t = 1, N(t) should be equal to 0.25 times the total population (0.25 * 250,000).
N(t = 1) = a(1 - e^(k * 1)) = 0.25 * 250,000
Plugging in a = 0, we get:
0(1 - e^k) = 0.25 * 250,000
Since 0 multiplied by any number is 0, this equation tells us nothing about the value of k.
Based on the given information, we can conclude that a = 0, but we cannot determine the value of k without further information. It seems there might be additional missing data or an error in the problem statement.