# Help creating a dif. eq. for a real life model?

Help creating a dif. eq. for a real life model.help???

A 30 year old woman accepts an engeineeering position with a starting salary of $30000 per year. Her salary S(t) increases exponentially, with $$S(t)=30e^{\frac{t}{20}$$ thousand dollars after t years. Meanwhile, 12% of her salary is deposited continuously in a retirement account, which accumulates interest at a continuous annual rate of 6%. a) Estimate dA in terms of dt to derive the differential equation satisfied by the amount A(t) in her retirement account after t years. Well, i am not any good at differential equations, but i am just trying to give a shot on my own, although i currently am in calculus I, so i am facing some difficulties from time to time absorbing these kind of problems, especially when they are wording ones that require us to derive a differential equation to describe the situation. I have tried many things on this problem, first i thought that the differential equation derived for measuring the concentration on a tank- as one pipe brings water in while the other pipes water out of tank- would work but i think it does not quite fit in this situation. so any hints on how to begin to derive a diff. eq for describing this situation?? thnx in advance Last edited: ## Answers and Replies Related Differential Equations News on Phys.org HallsofIvy Science Advisor Homework Helper A 30 year old woman accepts an engeineeering position with a starting salary of$30000 per year. Her salary S(t) increases exponentially, with $$S(t)=30e^{\frac{t}{20}$$ thousand dollars after t years. Meanwhile, 12% of her salary is deposited continuously in a retirement account, which accumulates interest at a continuous annual rate of 6%.
a) Estimate dA in terms of dt to derive the differential equation satisfied by the amount A(t) in her retirement account after t years.
A(t) increases in two ways:
1) She deposits 12% of her salary: 0.12 S= 0.12(30 et/20[/itex]= 3.6et/20.
2) interest, compounded continuously at 6% is added: A(t)e0.06t.
Those are the two ways in which the account changes and the rate of change is dA/dt.
Looks to me like dA/dt= e0.06tA(t)+ 3.6 et/20.

Well, i am not any good at differential equations, but i am just trying to give a shot on my own, although i currently am in calculus I, so i am facing some difficulties from time to time absorbing these kind of problems, especially when they are wording ones that require us to derive a differential equation to describe the situation.

I have tried many things on this problem, first i thought that the differential equation derived for measuring the concentration on a tank- as one pipe brings water in while the other pipes water out of tank- would work but i think it does not quite fit in this situation.

so any hints on how to begin to derive a diff. eq for describing this situation??

Shouldn't we take into consideration somewhere her starting salary, maybe you incorporated it somewhere but i just cannot se where? If not why would it be ok, even if we do not take it into consideration?
Also how did you derive A(t)e^0.06t ? In other words, how did you incorporate the 6% interest here?

Ben Niehoff
Gold Member
A 30 year old woman accepts an engeineeering position with a starting salary of $30000 per year. Geeze, I certainly hope not! That's an obscenely low salary for any 30-year-old, especially an engineer! HallsofIvy Science Advisor Homework Helper Shouldn't we take into consideration somewhere her starting salary, maybe you incorporated it somewhere but i just cannot se where? If not why would it be ok, even if we do not take it into consideration? Also how did you derive A(t)e^0.06t ? In other words, how did you incorporate the 6% interest here? The initial salary is already taken into account in the S(t)= 30et/20 although we should write it as 30000et/20. The 6% is the 0.06 in the exponential. Amount A, compounded continuously, at interest rate r, for t years yields Aert. HallsofIvy Science Advisor Homework Helper Geeze, I certainly hope not! That's an obscenely low salary for any 30-year-old, especially an engineer! First college teaching job I got, after I got my Ph.D, paid$12000 a year. And that was more than my father had ever earned in one year in his life.

Ben Niehoff
Gold Member
Yeah, but what decade was that?

Hell, even my first internship was \$18/hr (though I realize I'm a bit lucky there). :P

The 6% is the 0.06 in the exponential. Amount A, compounded continuously, at interest rate r, for t years yields Aert.
Where can i find some things to read on my own about this interest rate, because i do not actually know why amount A, compounded continuously, at interest rate r, for t years yeilds Aert??

Well, thank you HallsofIvy, i really appreciate it. I will come back from time to time with these kind of problems, untill i get used to them.

Thnx once more!