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8.) Find the amount A in an account after t years given the following conditions.

8.) Find the amount A in an account after t years given the following conditions.-example-1

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dA/dt = 0.02A


dA/A = 0.02dt


integrating left n right


lnA = 0.02t + C where C is a constant


at t=0, A=4000


C=ln4000


lnA = 0.02t + ln4000


0.02t = lnA-ln4000


t = ln(A/4000) / 0.02


t = 40ln(A/4000)


User Jamie Shepherd
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2 votes

This is a separable differential equation so can be written as


(dA)/(A)=0.02\,dt


Integrating, then taking antilogs, you have


\displaystyle\int{(dA)/(A)}=0.02\int{dt}\\\\\ln(A)+C=0.02t\\\\A=C\cdot e^(0.02t)


The exponential factor is 1 at t=0, so the multiplying constant is 4000.


A=4000e^(0.02t)

User TrueDub
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