The rate at which a rumor spreads through an Alpine village of 400 residents is jointly proportional to the number of residents who have heard it and the number who have not.
Initially, 5 residents heard the rumor, but 4 days later this number increased to 70 .
Find the number of people who will have heard the rumor after 2 wk.
To solve this problem, we can use a basic exponential growth equation. Let's define a variable "H" as the number of residents who have heard the rumor at a given time "t" in days.
Given that initially 5 residents heard the rumor, we have:
H(0) = 5
And 4 days later, the number of residents who have heard the rumor increased to 70, we have:
H(4) = 70
We are asked to find the number of people who will have heard the rumor after 2 weeks, which is equivalent to 14 days.
To find the constant of proportionality, let's use the fact that the rate at which the rumor spreads is jointly proportional to the number of residents who have heard it and the number who have not. Mathematically, we can express this as:
dH/dt = k * H * (400 - H)
Where "dH/dt" represents the rate of change of the number of residents who have heard the rumor, and "k" is the constant of proportionality.
To solve the differential equation, we will use the fact that the rate of change of H with respect to t is proportional to H times the difference between 400 and H.
Let's solve this differential equation:
Separating variables:
(dH / (H * (400 - H))) = k * dt
Integrating both sides:
∫ (dH / (H * (400 - H))) = ∫ k * dt
To simplify the integration, we can use partial fraction decomposition:
1 / (H * (400 - H)) = A / H + B / (400 - H)
Multiplying through by H * (400 - H):
1 = A * (400 - H) + B * H
Simplifying:
1 = 400 * A - A * H + B * H
Matching coefficients:
400 * A = 1 --> A = 1/400
-A + B = 0 --> B = A = 1/400
Therefore, the partial fraction decomposition is:
1 / (H * (400 - H)) = 1 / (400 * H) + 1 / (400 * (400 - H))
Now, integrating both sides:
∫ (1 / (H * (400 - H))) dH = ∫ (1/400H) + (1 / (400(400 - H))) dH
This gives us:
ln(H) - ln(400 - H) = (1/400) ln(H) - (1/400) ln(400 - H) + C
Simplifying:
ln(H) - ln(400 - H) - (1/400) ln(H) + (1/400) ln(400 - H) = C
Combining logarithms:
(399/400) ln(H) - (399/400) ln(400 - H) = C
Exponentiating both sides:
H^((399/400)) / (400 - H)^((399/400)) = e^C
Simplifying:
(H / (400 - H))^((399/400)) = e^C
Using the initial condition H(0) = 5:
(5/(400 - 5))^((399/400)) = e^C
Simplifying:
(5/395)^((399/400)) = e^C
Finding C:
C = ln((5/395)^((399/400))) = -0.9988
Substituting back into the equation:
(H / (400 - H))^((399/400)) = e^(-0.9988)
Raising both sides to the 400/399 power:
(H / (400 - H)) = e^(-0.9988)^(400/399)
Simplifying the exponent:
(H / (400 - H)) = e^(-1)
Multiplying both sides by (400 - H):
H = (400 - H) * e^(-1)
Expanding:
H = 400e^(-1) - He^(-1)
Moving H terms to one side:
H + He^(-1) = 400e^(-1)
Factoring out H:
H(1 + e^(-1)) = 400e^(-1)
Dividing both sides by (1+e^(-1)):
H = (400e^(-1)) / (1 + e^(-1))
Calculating the value:
H ≈ 146.90
Therefore, approximately 147 residents will have heard the rumor after 2 weeks.
To solve this problem, we can set up a differential equation to represent the rate at which the rumor spreads. Let's denote the number of residents who have heard the rumor at time t as H(t) and the number who have not heard it as N(t).
The given information tells us that initially, 5 residents heard the rumor, so H(0) = 5. Also, the total number of residents in the village is 400, so N(0) = 400 - H(0) = 395.
We are told that after 4 days, the number of people who heard the rumor increased to 70. So, we have the following information:
H(4) = 70 and N(4) = 400 - H(4).
Now, let's set up the differential equation to represent the rate at which the rumor spreads. We are given that the rate at which the rumor spreads is jointly proportional to the number of residents who have heard it and the number who have not. This can be written as:
dH/dt = k * H * N,
where k is a constant of proportionality. In other words, the rate of change of H with respect to time is proportional to H and N.
To solve this differential equation, we can separate the variables and integrate both sides:
dH/(H * N) = k * dt.
Integrating both sides will give us:
∫(1/(H * N)) dH = k * ∫dt.
The left integral can be solved as:
∫(1/(H * N)) dH = ln(H * N) + C1,
where C1 is a constant of integration.
The right integral can be solved as:
∫dt = t + C2,
where C2 is another constant of integration.
Now, substituting the integral results back into the equation, we have:
ln(H * N) + C1 = k * (t + C2).
We can rewrite this equation as:
ln(H * N) = k * t + (k * C2 - C1).
Exponentiating both sides, we get:
H * N = e^(k * t + (k * C2 - C1)).
Since H(t) represents the number of residents who have heard the rumor, and N(t) represents the number of residents who have not heard it, their product should be equal to the total number of residents in the village:
H * N = 400.
Therefore, we can solve for the constant of integration (k * C2 - C1) using the initial values H(0) = 5 and N(0) = 395:
5 * 395 = e^(k * 0 + (k * C2 - C1)).
Now, we have everything we need to find the solution for H(t) and N(t) at any given time t.
To find the number of people who will have heard the rumor after 2 weeks (14 days), we can substitute t = 14 into the equation:
H * N = e^(k * 14 + (k * C2 - C1)).
We can solve this equation to find the value of H(14).