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.

I found N'(t)=(ae^-(bx))+axe^-(bx)
and i figure a=6M*.05=300000, but i cannot find the %population after 5 days. any help would be awesome, Thanks in advance!

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

After 1 day, t=1, and we have

.05a = (1 - e^-k)
.05 = 1 - e^-k
0.95 = e^-k
k = -ln 0.95 = 0.0513
so,
N(t) = a(1 - e^-0.0513t)

So, by the end of day 5,

N(5) = a(1 - e^-.2565) = .226a = 22.6%

have heard the rumor

Well, calculating the percentage of the population who have heard the rumor after 5 days involves finding N(5) as a fraction of the total population (6 million).

First, let's find the value of k using the given information. We know that 5% of the population heard the rumor on the first day, so we can substitute t = 0 and N(t) = 0.05 * 6 million into the equation:

0.05 * 6 million = a(1 - e^(-k * 0))
300,000 = a

Now we can substitute the value of a back into the equation:

N(t) = a(1 - e^(-kt))
N(t) = 300,000 * (1 - e^(-kt))

Since we want to find N(5), substitute t = 5 into the equation:

N(5) = 300,000 * (1 - e^(-k * 5))

To calculate the percentage of the population who have heard the rumor after 5 days, we need to divide N(5) by the total population (6 million) and multiply by 100:

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

Please note that we still need to find the value of k, which can be done by using additional information or data.

To find the percentage of the population who have heard the rumor after 5 days, we need to evaluate the function N(t) at t = 5.

Given the equation N(t) = a(1 - e^(-kt)) and the information that 5% of the population heard the rumor on the first day, we can determine the value of a. Let's do the calculation:

a = 6 million * 0.05 = 300,000

Now, substitute a = 300,000 into the equation N(t):

N(t) = 300,000(1 - e^(-kt))

To find the percentage of the population who have heard the rumor after 5 days, substitute t = 5 into the equation:

N(5) = 300,000(1 - e^(-k*5))

Unfortunately, we don't have enough information to solve for the constant k. Without the value of k, we cannot determine the exact percentage of the population who have heard the rumor after 5 days.

If you have any additional information or constraints, please provide them, so we can proceed with the calculation.

To find the percentage of the population who have heard the rumor after 5 days, we need to evaluate N(5) in terms of the population as a percentage.

Given that N(t) = a(1−e^(-kt)), where N(t) represents the number of people who have heard the rumor by time t, and a represents the ultimate number of people who will eventually hear the rumor.

First, let's substitute the given values into the equation. We know that a = 6 million * 0.05, so a = 300,000.

Now, to find N(5), we need to calculate the number of people who have heard the rumor after 5 days. Substitute t = 5 into the equation:

N(5) = 300,000(1−e^(-k(5)))

To determine the percentage of the population who have heard the rumor after 5 days, we need to divide N(5) by the total population and multiply by 100:

Percentage = (N(5) / Population) * 100

Given that the total population is 6 million, we have:

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

To find N(5), we need to know the value of k, the rate at which the rumor spreads. Unfortunately, the question does not provide this information. Without the value of k, we cannot determine the specific percentage after 5 days.

You need to know the value of k in order to calculate the percentage of the population who have heard the rumor after 5 days.