The rate at which people learn about a secret is directly proportional to the number of people who know the secret. If two people know the secret on day 0, and 20 people know it on day 2, how long will it take for the whole school (4000 people) to know?
dy/dx = ky
dy/y = k dx
lny = kx + lnc
y(x) = c*e^(kx)
Now use
y(0) = 2
y(2) = 20
to find c and k
Then solve for x when y=4000
or, use the points to discover that y = 2*10^(x/2)
To solve this problem, let's first examine the given information. We are told that the rate at which people learn about a secret is directly proportional to the number of people who know the secret. In other words, the more people who know the secret, the faster it spreads.
We are given that on day 0, two people know the secret. On day 2, 20 people know it. This means that in two days, the number of people who have learned the secret increased by 20 - 2 = 18.
Now, let's try to find the rate of increase in the number of people who know the secret. We know that this rate is directly proportional to the number of people who already know the secret. Therefore, we can set up a proportion:
(Change in the number of people who know the secret) / (Change in time) = k
where k is the proportionality constant.
We have already calculated the change in the number of people who know the secret (18) and the change in time (2 days). Plugging these values into the proportion, we get:
(18) / (2) = k
Simplifying, we find that k = 9.
Now, let's use this information to find out how long it will take for the whole school (4000 people) to know the secret. We need to find the time it takes for the number of people who know the secret to increase from 2 to 4000.
Let's call the time it takes for this to happen "t". Using the proportionality constant we found earlier (k = 9), we can set up another proportion:
(Change in the number of people who know the secret) / (Change in time) = k
(4000 - 2) / t = 9
Simplifying, we have:
3998 / t = 9
To solve for t, we can multiply both sides by t:
3998 = 9t
Dividing both sides by 9:
444.22 = t
Therefore, it will take approximately 444.22 days for the whole school (4000 people) to know the secret.