Optimizing Savings Growth with Continuous Compounding and Depletion

[Alex Mamaev]

Homework Statement

Find the amount in a savings account after one year if the initial balance in the account was $1000, the interest is paid continuously into the account at a normal rate of 10% per annum (compounded continuously), and the account is being continuously depleted at the rate of y^2 per million dollars per year. the balance in the account after t years is y=y(t). How large can the account grow? How long will it take the account to grow to half of this maximum balance

Homework Equations

the differential equation which i think is correct is:
dy/dt= y/10 - y^2/1000000

The Attempt at a Solution

I solved the equation by separating and then doing partial fractions,
I got y= 100000Ce^(t/10)/(1+ke^(t/10))
with initial values this became 100000(e^(t/10))/(99+e^(t/10))
here is where i don't really know what to do. I took the derivative and that was always positive so I'm not sure how to find the maximum or if I've made an error.
Based on the initial differential equation, there should be an optimum value when y = 100000 but for some reason the function isn't defined there. Please help if possible really lost with this one, thanks.

 

[HallsofIvy]

No, you are not "really lost", you have just not realized a critical point: If the slope is always positive, then the function keeps increasing so the maximum occurs as to goes to infinity. It is easy to see that the limit, as t goes to infinity, is 100000. Now, what is t when the value is 50000?

 

[Alex Mamaev]

Omg thank you so much, i kept thinking it was an optimization problem, don't know why. I get it now:)