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fogvajarash
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Homework Statement
a. Assume that yo dollars are deposited into an account paying r percent compounded continuously. If withdrawals are at an annual rate of 200t dollars (assume these are continuous), find the amount in the account after T years.
b. Consider the special case if r = 10% and y0=$20000
c. When will the account be depleted if y0=$5000? Give your answer to the nearest month.
Homework Equations
The Attempt at a Solution
I've realized that the rate at which the account balance varies is the following:
dy/dt = ry - 200 (where r is the r percent rate, 0.10; and y the amount of money present)
However, when i try to obtain the differential equation, I keep getting that the amount of money present is the following:
y(T) = 200/r + (y0-200/r)erT
This would, mean that the function would never decrease in the case of $20000 and as well for $5000 (meaning it will never be depleted). However, I'm pretty sure that I'm wrong on this one. Could anyone please help me with this? My procedure:
1/(ry-200) dy = 1 dt (integrate both parts)
ln(ry-200) 1/r = t + M1
ln(ry-200) = rt + M2
M3ert=ry-200
y = M4ert + 200/r
Then, if y(0) = y0:
y0 - 200/r = M4
We then plug this result into our equation:
y = 200/r + (y0 - 200/r)erT
This corresponds to the equation I've been getting. Is my procedure done right?