Could u help me with this question.
The number, N, of people who have heard a rumor spread by mass media at time, t, in days is modelled by
N (t) = a / 1+ be^ -kt
a. If 50 people have heard the rumour initially and 300,000 people hear the rumour eventually, find a and b
b. If the rumour is initially spreading at the rate of 500 people per day, find k.
Thanks.
for a)
initially ---> t=0
so 50 = a/(1 + be^0)
50 = a/(1+b)
a=50 + 50b ----- (#1)
I will interpret "eventually" to mean t --> ∞
then e^(-large) ---> 0
and 300000=a/(1+0), so a=300000
putting this back in #1 gives b=5999
so N(t) = 300000/(1 + 5999e^-kt)^-1
for b)
N'(t) = -300000((1 + 5999e^-kt)^-2)(e^-kt)(-k)
"initially" the rate is 500 means
500 = -300000(1+5999)^-2 * (1)(-k)
..
..
I got k=10, check my arithmetic please
Sure, I can help you with this question.
a) To find the values of a and b, we will use the given information.
Initially, at t = 0, the number of people who have heard the rumor is 50. We can substitute this value in the equation:
N(t) = a / (1 + be^(-kt))
50 = a / (1 + b * e^(0))
50 = a / (1 + b)
So, we have the equation a = 50 + 50b. (Equation #1)
Next, we interpret "eventually" to mean as t approaches infinity. In this case, e^(-kt) approaches 0, making the denominator (1 + be^(-kt)) equal to 1.
300,000 = a / (1 + b * e^(-kt))
300,000 = a / (1 + 0)
a = 300,000
Now, substitute the value of a = 300,000 in Equation #1:
300,000 = 50 + 50b
250,000 = 50b
b = 5999
So, the values of a and b are a = 300,000 and b = 5999.
b) To find the value of k, we need to find the rate at which the rumor is spreading initially, which is given as 500 people per day.
Differentiating the equation N(t) with respect to t, we have:
N'(t) = -a * b * e^(-kt) / (1 + be^(-kt))^2 * (-k)
Plugging in the initial rate of 500:
500 = -300,000 * (1 + 5999)^(-2) * 1 * (-k)
Simplifying this equation, we get:
500 = -300,000 * (1 + 5999)^(-2) * (-k)
Solving for k:
k = -500 / (-300,000 * (1 + 5999)^(-2))
Evaluating this expression, we find k = 10.
Therefore, the value of k is 10.