Fads come and go quickly. (Denition: a fad is something (such as an interest or

ID: 1888912 • Letter: F

Question

Fads come and go quickly. (Denition: a fad is something (such as an interest or fashion) that is

very popular for a short time). Suppose that, in general, the rate that people become interested or

disinterested in any particular fad is jointly proportional to the number of people interested in the

fad and to how long till the peak of the fad. If the peak of a particular fad is 3 months after it starts,

and initially 5 thousand people are interested in it, and people are initially becoming interested at a

rate of 30 thousand per month, how many people will be interested in the fad at its peak?

Please show me your work

Let rate be r, number of people interersted = n, time to peak = t

Now, we have

r = dn / dt

also, r is directly proportional to number of people interested and time to peak

r = knt ------> k is a constant

Initially, r = 30000, n=5000, t =3

k = 2

r = 2nt

2nt = dn/dt

dn / n = 2tdt

Integrating we get, n = cet²

Plugging the end-values,

N (n at peak) = n * et² = 5000 * e9 = 40515419

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