We have two pieces of information that we can use to find a and k:
1. N = a(1−e^(−k*t)) and N = 5,000,000
2. 10% of 5,000,000 (or 500,000 people) heard the rumour on the first day. This means that N(1) - N(0) = 500,000.
Substituting N = 5,000,000 into the formula gives:
5,000,000 = a(1−e^(−k*t))
Substituting t = 1 and N(1) - N(0) = 500,000 gives:
500,000 = a(1−e^(−k*1))
Simplifying the second equation:
500,000 = a(1−e^(−k))
500,000/a = 1−e^(−k)
e^(−k) = 1 - 500,000/a
−k = ln(1 - 500,000/a)
k = -ln(1 - 500,000/a)
We can now solve for a and k by substituting the expression for k into the first equation:
5,000,000 = a(1−e^(ln(1 - 500,000/a)*t))
5,000,000 = a(1 - (1 - 500,000/a)^t)
Now, we use the fact that we want to find the percentage of people who heard the rumour on the second day (t=2).
N(2) - N(1) is the number of people who heard the rumour on the second day.
N(2) - N(1) = a(1 - (1 - 500,000/a)^2) - a(1 - (1 - 500,000/a))
Simplifying:
N(2) - N(1) = a(1 - (1 - 500,000/a))((1 - 500,000/a) + 1)
N(2) - N(1) = 500,000(500,000/a)(1 - 500,000/a)
To find the percentage of people who heard the rumour on the second day, we divide N(2) - N(1) by the total number of people who heard the rumour:
(N(2) - N(1))/N = (500,000(500,000/a)(1 - 500,000/a))/5,000,000
Simplifying:
(N(2) - N(1))/N = 0.0976 or 9.76%
Therefore, the percentage of people who heard the rumour on the second day is 9.76%.
The number, N, of people who eventually hear a rumour spread by the mass media by time t, in days, is given by the formula N = a(1−e^(−k*t)) for positive numbers a and k. Suppose 5, 000, 000 people hear a rumour eventually. Further, 10% of them heard the rumour in the first day. Find the values of a and k. What percentage of people heard the rumour on the second day?
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