Very difficult exponential applications question, help.

2008-01-11 9:09 pm

A particular town population, 20% of the residents heard radio announcement about local political scandal. After hours f(t) people had heard the scandal, here f(t)=A/(1+Be^-Akt). If 50% of the population heard the scandal after 1 hour, how long was it until 80% of the population heard about it?

更新1:

Exponential functions are useful for dealing with radioactive decay. The half-life of radioactive element is the time takes a given quantity to decay to one half of its original mass. The half-life depends on the substance and not depend on the sample size.

更新2:

Let suppose the function M(t)=c(1/2)^t/h give the mass at time( t0, where h is half-life and c is the left original mass of element.

更新3:

我還是不完全明白.

回答 (1)

TszFung

2008-01-11 9:38 pm

✔ 最佳答案

f(t)=A/(1+Be^-Akt)
when t tends to infinity, f(t) tends to A
thus the total population is A

Initially, f(0) = A/5
A / (1+B) = A/5
B = 4

f(1) = A/2
A / [1+4e^(-Ak)] = A/2
-Ak = -ln4

Therefore,
f(t)
=A/(1+Be^-Akt)
= A / [1+4^(-tln4)]
= A / [1+4^(1-t)]

For the equation f(T) = 4A/5
T = 2

So, 80% of the population heard about it after 2 hours

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