Expontential Growth

samantha0417

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Sep 24, 2006
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an employer determines that the daily output of a worker for t weeks is Q(t)= 120 - Ae^(-kt) units. Initially, the worker can produce 30 units a day and after 8 weeks he can produce 80 units a day. how many units can the worker produce a day after 4 weeks?
 
Hello, Samantha!

An employer determines that the daily output of a worker for \(\displaystyle t\) weeks is: \(\displaystyle Q(t)\:= \:120\,-\,Ae^{-kt}\) units.
Initially, the worker can produce 30 units a day and after 8 weeks he can produce 80 units a day.
How many units can the worker produce a day after 4 weeks?

The two conditions are given so we can determine \(\displaystyle A\) and \(\displaystyle k\).

At \(\displaystyle t\,=\,0,\:Q\,=\,30\)
. . \(\displaystyle 120\,-\,Ae^0\:=\:30\;\;\Rightarrow\;\;A\,=\,90\)

The function (so far) is: \(\displaystyle \:Q(t)\:=\:120\,-\,90e^{-kt}\)


At \(\displaystyle t\,=\,8,\:Q\,=\,80\)
. . \(\displaystyle 120\,-\,90e^{-8k}\:=\:80\;\;\Rightarrow\;\;90e^{-8k}\:=\:40\;\;\Rightarrow\;\;e^{-8k}\:=\:\frac{4}{9}\)

. . \(\displaystyle \,-8k\:=\:\ln\left(\frac{4}{9}\right)\;\;\Rightarrow\;\;k\:=\:-\frac{\ln\left(\frac{4}{9}\right)}{8} \:\approx\:0.101\)

Hence, the function is: \(\displaystyle \,Q(t)\;=\;120\,-\,90e^{-0.101t}\)


When \(\displaystyle t\,=\,4:\:Q(4)\;=\;120\,-\,90e^{-0.101(4)} \:\approx\:\fbox{60\text{ units.}}\)

 
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