Suppose you deposit 8500 dollars into a savings account earning 5 percent annual interest compounded continuously. To pay for all your music downloads, each year you withdraw $900 in a continuous way.
Let A(t) represent the amount of money in your savings account t years after your initial deposit.
(A) Write the DE model for the time rate of change of money in the account. Also state the initial condition.
dA/dt=?
A(0)=?
(B) Solve the IVP to find the amount of money in the account as a function of time.
A(t)=?
(C) When will your money run out?
t=?
Best Answer
As per the notions of differential coefficient of continuous derivatives ( compounding takes place each millisecond !),with annual music download payment $M$ deducted.
$$ \frac{dA}{dt} = r\cdot A- M $$
Integrating $$ \log( r A - M) = r \, t + C $$
At start using boundary condition $$ t= 0 , A =P $$
$$ log \,( r p - M) = C $$
$$ log\, \frac{r \, A - M }{r \,P - M} =r\, t $$
$$ A = ( P- M/r) e ^{r\, t} + M/r $$
$ P = 8500 ; r = 0.05 ; M=900; t = $ no of years.
Bank balance using calculus and influence of each constant can be graphed.