The number, N, of people who have heard a rumor spread by mass media by time, t, is given by N(t)=a(1−e−kt). There are 6 million people in the population, who hear the rumor eventually. If 5% of them heard it on the first day, find the percentage of the population who have heard the rumor after 5 days.

Question

The number, N, of people who have heard a rumor spread by mass media by time, t, is given by N(t)=a(1−e−kt). There are 6 million people in the population, who hear the rumor eventually. If 5% of them heard it on the first day, find the percentage of the population who have heard the rumor after 5 days.

Answer 1

To find the percentage of the population who have heard the rumor after 5 days, we need to substitute the value of t as 5 in the equation N(t) = a(1 - e^(-kt)) and then calculate it as a percentage of the total population.

Given that 5% of the population heard the rumor on the first day, we can set up the equation using the information provided:

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

We know that N(t) = 0.05a, since 5% of the population represents the number of people who heard the rumor on the first day.

0.05a = a(1 - e^(-k(5)))

Now, we can simplify the equation by canceling out 'a' from both sides:

0.05 = 1 - e^(-5k)

Rearranging the equation, we get:

0.05 - 1 = -e^(-5k)

-0.95 = -e^(-5k)

Since the equations on both sides of the equality are equal for any value of k, we can take the natural logarithm (ln) of both sides to solve for k:

ln(-0.95) = ln(-e^(-5k))

ln(-0.95) = -5k

Dividing by -5 to isolate 'k', we get:

k = ln(-0.95) / -5

Using a calculator or a computational tool, we find that k ≈ 0.04155.

Now that we have the value of 'k', we can find the percentage of the population who have heard the rumor after 5 days.

N(5) = a(1 - e^(-k(5)))

We are given that the total population is 6 million, so we can substitute 'a' as 6 million:

N(5) = 6 million(1 - e^(-0.04155(5)))

N(5) ≈ 6 million(1 - e^(-0.20775))

Using a calculator or a computational tool, we can evaluate the expression on the right-hand side:

N(5) ≈ 6 million(1 - 0.799037)

N(5) ≈ 6 million(0.200963)

N(5) ≈ 1,205,778

To find the percentage, we divide N(5) by the total population and multiply by 100:

Percentage = (N(5) / 6 million) * 100

Percentage = (1,205,778 / 6,000,000) * 100

Percentage ≈ 20.09%

Therefore, approximately 20.09% of the population have heard the rumor after 5 days.