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.

what do I use for a and k in this situation? i tried taking the derivative but no answers i come up make sense based on the problem.

Given that the number of people who have heard the rumor by time t is given by N(t) = a(1 - e^(-kt)), where N(t) is the number of people who have heard the rumor, a is the total population, k is a constant, and t is the time in days.

We are given that eventually, all 6 million people in the population will hear the rumor. So, when t approaches infinity, N(t) will be equal to 6 million.

If 5% of the population heard the rumor on the first day, we can say that N(1) = 0.05 * a.

We need to find the percentage of the population who have heard the rumor after 5 days, or N(5)/a * 100%.

To find the value of k, we can use the fact that N(t) approaches 6 million as t approaches infinity. Hence, we can say that lim(t→∞) N(t) = 6 million.

Substituting the formula, we have:
lim(t→∞) a(1 - e^(-kt)) = 6 million

Dividing both sides by a, we get:
lim(t→∞) (1 - e^(-kt)) = 6 million / a

Since the limit as t approaches infinity of e^(-kt) is 0, we have:
1 = 6 million / a

Simplifying, we find:
a = 6 million

Now, we can substitute a = 6 million in N(1) = 0.05 * a, giving us:
N(1) = 0.05 * 6 million
N(1) = 0.3 million

To find k, we can substitute N(1) and a into the formula N(t) = a(1 - e^(-kt)):
0.3 million = 6 million * (1 - e^(-k * 1))

Dividing both sides by 6 million, we get:
0.3 / 6 = 1 - e^(-k)

Simplifying, we have:
0.05 = 1 - e^(-k)

Subtracting 1 from both sides, we find:
-0.95 = -e^(-k)

Taking the natural logarithm of both sides, we have:
ln(0.95) = -k

Using a calculator to evaluate ln(0.95), we find:
k ≈ -0.051293

Now, we can find N(5) using the formula N(t) = a(1 - e^(-kt)):
N(5) = 6 million * (1 - e^(-(-0.051293 * 5)))

Simplifying, we get:
N(5) ≈ 6 million * (1 - e^(0.256465))
N(5) ≈ 6 million * (1 - 0.774111)
N(5) ≈ 6 million * 0.225889
N(5) ≈ 1.355334 million

To find the percentage of the population who have heard the rumor after 5 days, we divide N(5) by a and multiply by 100%:
Percentage = (N(5) / a) * 100%
Percentage = (1.355334 million / 6 million) * 100%
Percentage ≈ 22.59%

Therefore, approximately 22.59% of the population will have heard the rumor after 5 days.

To find the percentage of the population who have heard the rumor after 5 days, we need to substitute the given values into the equation N(t) = a(1−e^(-kt)) and solve for N(5).

Given:
N(t) = a(1−e^(-kt))
N(0) = 0.05a (5% of the population heard the rumor on the first day)
N(∞) = 6 million (the total population, eventually, heard the rumor)
t = 5 (5 days)

We can use the information that N(0) = 0.05a to solve for the value of a. 0.05a represents the number of people who heard the rumor on the first day, which is 5% of the population. Therefore, a = 0.05 * 6 million.

Substituting the values, we have:
N(t) = (0.05 * 6 million)(1−e^(-kt))

Now, we can solve for N(5):
N(5) = (0.05 * 6 million)(1−e^(-k * 5))

To find 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 to get the percentage.

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

By substituting the value of N(5) into the equation and performing the calculations, we can find the answer.