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

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 explain all work

Explanation / Answer

N(t) = number of interested people at time, t, where t is the number of months until the peak.

t(at peak) = 0 ---> because we no longer have to wait for the peak.

N'(t) = rate at which people become interested

It will be some quadratic about x=0. y is in thousands. The fad started at t=-3 (3 months before the peak)

y(t) = at2 + bt + c   ---> y(t) = at2 - c (because the b-term vanishes since this curve is symetrical about the y-axis)

y(t) = at2 - c

y(-3) = 5 = a(-3)2 - c ---> 5 = 9a - c

y'(t) = 2at = rate at which people interested

y'(-3) = 2a(-3) = 30

-6a = 30

a = -5

5 = 9a - c

5 = 9(-5) - c

5 = -45 - c

c = -50

c is the y-intercept and, because the curve is symetrical about the y-axis, it is also the y-coordinate of the peak.

Max interest at peak = 50000 people

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