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

[sutupidmath]

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

 

[HallsofIvy]

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)e^0.06t.
Those are the two ways in which the account changes and the rate of change is dA/dt.
Looks to me like dA/dt= e^0.06tA(t)+ 3.6 e^t/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??

thnx in advance

 

[sutupidmath]

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]

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]

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)= 30e^t/20 although we should write it as 30000e^t/20.

The 6% is the 0.06 in the exponential. Amount A, compounded continuously, at interest rate r, for t years yields Ae^rt.

 

[HallsofIvy]

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]

Yeah, but what decade was that?

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

 

[sutupidmath]

The 6% is the 0.06 in the exponential. Amount A, compounded continuously, at interest rate r, for t years yields Ae^rt.

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 Ae^rt??

 

[HallsofIvy]

Wikipedia is always a good place to start:
http://en.wikipedia.org/wiki/Compound_interest

 

[sutupidmath]

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

Thnx once more!