How can I solve this DiffEQ problem involving retirement savings and salary increases?

[mailman85]

I am having a lot of trouble solving this problem. I don't even know where to start. Any help would be greatly appreciated.

A 30 year old woman accepts an engineering position with a starting salary of $30000 per year. Her salary S(t) increases exponentially with S(t)=30e^(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 change(A) in terms of change(t) to derive the differential equation satisfied by the amount A(t) in her retirement account after t years. b) Compute A(40), the amount available for her retirement at age 70.

 

[HallsofIvy]

A, the amount of money in the retirement account, changes in two ways: 1) she contributes money each year and 2) it earns interest.
The amount of money she contributes is 12% of her salary: 0.12S
and the interest is 6% of the amount in the account: 0.06A.

[DELTA]A= 0.12S+ 0.06A= 0.12(30exp(t/20)+ 0.06A

That's the amount each year. If [DELTA]t is a portion of the year, then each of these would be multiplied by [DELTA]t:

[DELTA]A= (0.12S+ 0.06A)[DELTA]t
= (0.12(30exp(t/20)+ 0.06A)[DELTA]t
[DELTA]A/[DELTA]t = 3.6 exp(t/20)+ 0.06A.

The differential equation is dA/dT= 3.6 exp(t/20)+ 0.06A or
dA/dt- 0.06A= 3.6 exp(t/20), a relatively straight-forward first order, non-homogeneous, linear equation with constant coefficients.

Assuming she started this retirement account when she started the job, then A(0)= 0 is the intial condition.

(I get that, after 40 years, her retirement account contains
1308.28330 thousand dollars or $1,308,283.30.) Might be enough to retire on!

 

[tacman]

Firstly, it is important to understand the given information and what is being asked. The problem involves a 30-year-old woman who starts with a salary of $30,000 per year and her salary increases exponentially over time. Meanwhile, 12% of her salary is continuously deposited into a retirement account, which accumulates interest at a rate of 6% per year. The goal is to find the differential equation for the amount in her retirement account after t years and to compute the amount available for her retirement at age 70.

To start solving this problem, we need to understand the concept of continuous compounding and how it relates to exponential growth. In this case, the woman's salary is increasing exponentially with a growth rate of 5% per year (since S(t) = 30e^(t/20) and 1/20 = 0.05). This means that her salary is increasing by 5% every year.

Next, we need to consider the amount being deposited into the retirement account. Since 12% of her salary is continuously deposited, we can represent this as 0.12S(t). This means that the amount deposited into the retirement account is also increasing exponentially with a growth rate of 5% per year.

Now, we can use the formula for continuous compounding to calculate the amount in the retirement account after t years, which is given by A(t) = A(0)e^(rt), where A(0) is the initial amount, r is the annual interest rate, and t is the time in years. In this case, A(0) is 0 (since no initial amount is given), r is 6% (since the interest rate is 6% per year), and t is the time in years.

Therefore, the differential equation for the amount in the retirement account after t years can be written as dA/dt = 0.12S(t) = 0.12(30e^(t/20)) = 3.6e^(t/20).

To compute A(40), the amount available for her retirement at age 70, we simply plug in t = 40 into the equation for A(t) and solve for A(40). This gives us A(40) = 0.12(30e^(40/20)) = $2,011,706.67.

In conclusion, to solve this problem, we used the concept of