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.
"eventually means as t→∞
so, assuming k<0, that means a = 250000
So now you know that
250000(1 - e^k) = 250000/4
1-e^k = 1/4
e^k = 3/4
k = ln 3/4
N(t) = 250000(1 - e^(ln 3/4 * t))
...
But, since e^(ln 3/4) = 3/4, that gives
N(t) = 250000(1 - (3/4)^t)
so N has a (3/4)-life of 1 day
1243
To find the values of "a" and "k" in the equation N(t) = a(1 - e^(kt)), we can use the given information that 25% of the population heard the rumor on the first day, and eventually 250,000 people heard the rumor.
Let's analyze the equation step by step:
1. On the first day, t = 0. Therefore, N(0) = a(1 - e^(k*0)).
Since we know that 25% of the population heard the rumor on the first day, N(0) = 0.25(250,000) = 62,500.
Plugging in these values, we get 62,500 = a(1 - e^0) = a.
2. Eventually, 250,000 people heard the rumor. Therefore, as t approaches infinity, N(t) approaches 250,000. This means N(t) = 250,000 when t → ∞.
Using the equation, we have 250,000 = a(1 - e^(k*∞)) = a(1 - e^∞).
Notice that e^∞ approaches infinity, so 1 - e^∞ approaches 1 - ∞ = -∞.
Therefore, we can't directly solve for k using this information.
In summary:
- The value of "a" is 62,500.
- The value of "k" can't be determined solely based on the given information.
Note: To find the value of "k," more information is needed, such as the rate at which the rumor spreads or the time it takes for a certain percentage of the population to hear the rumor.
To find the values of a and k, we can use the given information about the population and the percentage of people who heard the rumor on the first day.
Let's start by analyzing the given equation:
N(t) = a(1 - e^(kt))
Here, N(t) represents the number of people who have heard the rumor at time t, a is a constant representing the maximum number of people who will eventually hear the rumor, k is a constant that determines the rate of spread, and e is Euler's number, approximately equal to 2.71828.
We are given that eventually, 250,000 people will hear the rumor. This means that when t approaches infinity, N(t) approaches 250,000:
lim(t→∞) N(t) = 250,000
Substituting this into the equation, we get:
250,000 = a(1 - e^(k∞))
Since e raised to any power remains finite, e^(k∞) approaches either 0 or 1, depending on the value of k. So, we can rewrite the equation as:
250,000 = a(1 - 0) or 250,000 = a(1 - 1)
This simplifies to:
250,000 = a or 250,000 = 0
Since we know that 250,000 is the value for a, we can conclude that a = 250,000.
Now, let's use the other given information: 25 percent of the population heard the rumor on the first day. We can express this mathematically as:
N(1) = a(1 - e^(k * 1))
Since N(1) represents the number of people who heard the rumor on the first day, and we know it is 25% of the population, we have:
N(1) = 0.25 * 250,000
Now we can substitute these values into the equation and solve for k:
0.25 * 250,000 = 250,000(1 - e^k)
Dividing both sides of the equation by 250,000:
0.25 = 1 - e^k
Rearranging the equation:
e^k = 1 - 0.25
e^k = 0.75
To solve for k, we take the natural logarithm (ln) of both sides:
ln(e^k) = ln(0.75)
k = ln(0.75)
Using a calculator, we can find that k is approximately -0.2877.
Therefore, the values of a and k are a = 250,000 and k ≈ -0.2877.