Finding the Value of a Compounded Annuity Fund

[alane1994]

A high school mathematics teacher puts \(\$\)2000 into an annuity fund and then contributes \(\$\)1800 per year into the fund for the next 30 years by making small weekly contributions. (We assume weekly contributions are close enough to continuous deposits so that we may use a differential equation model.) The fund grows at a rate of 7.5% per year.

(a) Write a differential equation that models the growth of this fund using \(m(t)\) for the amount of money present in the fund.
(b) How much money will be in the fund after 30 years according to this model.

I feel confident that I can solve (b)

I am confused because 7.5% isn't an interest rate or anything...

------EDIT------

When I try and put it into

\(Pe^{rt}\)

It doesn't come out right, I am completely baffled as to how to proceed.

 

[MarkFL]

I would take the 7.5% annual growth to be due to interest. So, we have two things contributing to the growth of the fund...the weekly contributions (which we are told to model as continuous) resulting in an annual growth and the annual growth due to interest. We are also given an initial value. Can you put all of this together to get an IVP that models the situation?

 

[alane1994]

Hmm...

For a continuous growth rate, would we have?

\(\displaystyle \lim\limits_{n\rightarrow\infty}(1+\frac{0.75}{n})^{n}\)

1.07788
so r = .07788 perhaps?

 

[MarkFL]

What you want to do is write an IVP consisting of an ODE that describes how $m(t)$ (in dollars) changes with time $t$ (in years), and the initial amount present in the account:

\(\displaystyle \frac{dm}{dt}=\text{annual contributions}+\text{annual growth from interest earned}\) where \(\displaystyle m(0)=\text{initial investment}\)

We are told the annual contributions total \$1800, and the initial investment is \$2000. Now, the annual growth from interest will be a function of $m(t)$...:D

 

[blue_raver22]

Hello,

I can provide you with some guidance on how to approach this problem.

(a) The differential equation that models the growth of this fund can be written as:

\(\frac{dm}{dt} = 0.075m + 1800\)

Where \(m(t)\) represents the amount of money in the fund at time \(t\) and 0.075 is the annual growth rate of 7.5% converted to a decimal.

(b) To solve for the amount of money in the fund after 30 years, we can use the formula for compound interest:

\(A = P(1+r)^n\)

Where \(A\) is the final amount, \(P\) is the initial amount, \(r\) is the annual growth rate, and \(n\) is the number of compounding periods. Since the contributions are made weekly, we can convert 30 years to 1560 weeks.

Plugging in the values, we get:

\(A = (2000 + 1800)(1+0.075)^{1560} \approx \$1,235,075.66\)

So, according to this model, after 30 years, there will be approximately $1,235,075.66 in the fund.

I understand your confusion about the 7.5% growth rate. In this context, it represents the annual interest rate that the fund is growing at. In other words, for every $100 in the fund, it will grow by $7.5 in one year.

I hope this helps you understand and solve the problem. Let me know if you have any further questions.