Question

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)

Answer 1

since it involves a rate of change, it can be modeled by a diffy q

Answer 2

spose a barrel of water drains at a constant rate of 1800 liters per week into another container that already has 2000 liters of water in it. etc ...

Answer 3

lol, 1800 liters per year that is

Answer 4

Well, I suppose what is throwing me is the grows at a rate of 7.5%/year

I mean without that, this would simply be

\(m(t)=2000+1800x\)
\(\text{where}~x~\text{is in years}\)

Answer 5

yeah, the usual conitnuous rate is what: Pe^(rt)

Answer 6

i cant tell if that 7.5% is an interest rate or not ....

Answer 7

I don't know either... it just says that it grows at that rate....

Answer 8

2000 e^(.075) = 2156 and that not a 1800 increase is it

Answer 9

Nope... so.... I am just very confused by this problem. How on earth?

Answer 10

I'm going to ponder this for a moment.

Answer 11

1.075 = e^r
ln(1.075) = r

is still 150 a year

Answer 12

3800 = 2000 e^r

1.9 = e^r

r = ln 1.9 is equal to 64.79% ... yeah, this is odd

Answer 13

This is certainly headache inducing.

Answer 14

$2000 is put into an annuity fund
$1800 per year added for the next 30 years
divided into small weekly contributions.

The fund grows at a rate of 7.5% per year.

well that didnt help me out any ....

Answer 15

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.

..... so.....

Answer 16

7.5% is interest at the end of the year .....

lim(n to inf) (1+.075/n)^n might be what we are after for a continuous groth rate

Answer 17

1.07788

so r = .07788 maybe?

Answer 18

i got nothing concrete coming to mind, and i have to get back to work :/

Answer 19

But what I don't get is how the heck to make a differential equation problem.
And thanks very much for your help! Much appreciated.

Answer 20

modeling diffy qs has never been my strong point

Answer 21

http://tutorial.math.lamar.edu/Classes/DE/Modeling.aspx

this might help

Answer 22

Yep, I am going to do what man has done since the beginning of time... give up. It is the noble and just thing to do. I am seriously ready to smash a computer right now.

Answer 23

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:

dm/dt=annual contributions+annual growth from interest earned where m(0)=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)

what would the m(t) be?

Answer 24

Part of me really wants to cheat this question by answering
\[\Large 2000 + 1800s_{\overline{30|}}\]

Answer 25

Where
\[\LARGE s_{\overline{30|}}=\frac{1.075^{30}-1}{0.075}\]

Answer 26

How in tarnation......

Answer 27

An annuity... I remember it has a formula to it.
If i is the interest rate per year, and an amount x is deposited per year, then it accumulates to the value
\[\Large x\cdot s_{\overline{n|}}\] after n years.

Answer 28

However, it can't be this simple... hang on.

Answer 29

I am really rather confuzzled by this.

What is,

\(\Huge{s_{\over{n}|}}\)

Answer 30

So sorry.
\[\huge s_{\overline{n|}}= \frac{(1+\color{green}i)^n-1}{\color{green}i}\]
where i is the interest rate per year.

It's the accumulated value at the end of the nth year, if you deposit 1 per year for n years. (annuity)

Answer 31

I give up, I just give up.
You see how long ago this was asked....

Answer 32

It's not like you to be giving up :D

Answer 33

That is what has me concerned :P

Answer 34

I'm having a hard time with this.. I already have access to the dogma that is the theory of interest and am therefore having a hard time proceeding normally.... heaven help us XD

Answer 35

Heaven has already forsaken this problem.... that means that there is only one place left to turn, and it is very warm.

Answer 36

What's that? The oven? :3

Answer 37

Yes Terence... the oven...

Well, I have to head off to psych class now. Wish me luck

Answer 38

^_^