The number of people that hear a rumor follows logistic growth. In a school of 1500 students, 5 students start a rumor. After 2 hours, 120 students have heard about the rumor.
Recall: dy/dx=ky(1-Y/L) and y=L/(1+be^(-kt))
I found the logistic growth equation to be 1500/(1+299e^-1.628548519t)
What is the rate of growth when the rumor is spreading the fastest?
y = 1/(1+be^kt)
y' = kbe^kt/(1+be^kt)^2
y' will be greatest when y" = 0
y" = k^2be^kt(3e^kt-1)/(1+be^kt)^3
so, that will be when t = -1/k ln3
In this case, when t = 1.48
To find the rate of growth when the rumor is spreading the fastest, we need to find the maximum value of dy/dt (rate of growth) by finding the critical points of the logistic growth equation.
1. Start by finding the derivative of the logistic growth equation with respect to time (t), dy/dt:
dy/dt = d/dt (1500/(1+299e^-1.628548519t))
2. Apply the quotient rule to find the derivative:
dy/dt = (0 - 1500 * d/dt(1+299e^-1.628548519t))/(1+299e^-1.628548519t)^2
3. Simplify:
dy/dt = -1500 * (-299 * e^-1.628548519t) / (1+299e^-1.628548519t)^2
dy/dt = (448500e^-1.628548519t)/(1+299e^-1.628548519t)^2
4. To find the critical points, set dy/dt equal to zero and solve for t:
(448500e^-1.628548519t)/(1+299e^-1.628548519t)^2 = 0
448500e^-1.628548519t = 0 (Since the denominator is always positive)
e^-1.628548519t = 0 (Divided both sides by 448500, which is non-zero)
e^-1.628548519t = 0 is not solvable since the exponential function never equals zero.
Therefore, there are no critical points for dy/dt, meaning the rate of growth is always changing and there is no single point where the rumor is spreading the fastest. The rate of growth is continuously increasing until it reaches a maximum value but never stops changing.
To find the rate of growth when the rumor is spreading the fastest, we need to find the value of t at which the derivative of the logistic growth equation is maximum.
Recall that the derivative of y with respect to t (dy/dt) represents the rate of growth. Therefore, we need to find the value of t that maximizes dy/dt.
Let's differentiate the logistic growth equation with respect to t:
dy/dt = d/dt [1500/(1 + 299e^(-1.628548519t))]
To find the maximum, we need to set d(dy/dt)/dt = 0 and solve for t.
d(dy/dt)/dt = d^2y/dt^2 = 0
Now, differentiate dy/dt with respect to t again:
d^2y/dt^2 = d/dt [d(dy/dt)/dt]
After differentiating, we can set d^2y/dt^2 = 0 and solve for t:
d^2y/dt^2 = d/dt [d(dy/dt)/dt] = 0
Solving this equation will give us the value of t at which the rumor is spreading the fastest.
Alternatively, if you have the logistic growth function in the form of y = L/(1 + be^(-kt)), you can find the value of t for the maximum rate of growth by finding the value of t at which dy/dt is maximum.
To find dy/dt, differentiate the logistic growth function with respect to t:
dy/dt = d/dt [L/(1 + be^(-kt))]
Then, find the value of t that maximizes dy/dt by setting d(dy/dt)/dt = 0 and solving for t.
d(dy/dt)/dt = d^2y/dt^2 = 0
Solving this equation will give us the value of t at which the rumor is spreading the fastest.