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### Income from Winning the Lottery

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Date: 7/30/96 at 14:16:13
From: Anonymous
Subject: Winning the Lottery

How much money would I need to win in the lottery to have an income of
\$75,000.00 a year until I die?  That should be in about 30 more years.
```

```
Date: 7/31/96 at 12:58:44
From: Doctor Paul
Subject: Re: Winning the Lottery

So let's say you win x dollars and you don't want to keep it in the
bank. You just keep it under your bed or in a money vault like Scrooge
McDuck... You would not be earning any interest so you would need
75000 * 30 years = 2.25 million dollars.

However, If you decided to put the money in a bank and earn a pretty
meager 5% interest then you would need much less money b/c you would
be earning some money along the way due to interest.  We'll assume you
don't lose any of your dividends to Uncle Sam.  In this particular
case, we have a Differential Equation to solve.  Let's do it:

dY
-- = [ .05 * Y(t) ]  -  75,000
dt

Let's separate variables by cross multiplying:

We will solve this Differential Equation for Y(t), which is the amount
you have in the bank at time t.

dY
------------  =  dt
(.05Y - 75000)

Integrate the left side dY and integrate the right side dt.  Don't ask
why it works... it just does.  The proof is not worth the time it
would take to show it to you.

20 ln | .05Y - 75000 | = t + C           where C is an arbitrary
constant

divide both sides by 20 and exponientate both sides:

|.05Y - 75000| = e^(.05t + .05C)   If C is a constant, then so it .05C
= e^.05t * e^C   Now if C is a constant, then so is e^c
Let's rename it 'A'

| .05Y - 75000 | = A*e^(.05t)

Now we have to get rid of the absolute value signs.  Note that if
dY/dt is negative (which it always is in this case) then

| .05Y - 75000 | = 75000 - .05Y

Make the substitution:

75000 - .05Y = A*e^(.05t)

solve for Y:

-.05*Y = -75000 + A*e^(.05t)
Y = 1,500,000 - B*e^(.05t)  since A is a constant, so is A/.05
Rename it 'B'

Let's use an initial condition to solve for B.

We know that at t = 30 years that Y =0

Plug that in:

0 = 1500000 - B * e^(.05 * 30)

B = 334695

so Y(t) = 1500000 - 334695 * e^(.05 * t)

let's find out what Y(0) is.  That is how much money you need to start
with.

y(0) = 1500000 - 334695 = 1165305

So you would only need 1.16 million dollars if you made 5% interest.
If you invested in stocks or mutual funds (as most people with a lot
of money do) you would probably end up making 14 or 15 percent
interest and you would need to win even less money initially.

-Doctor Paul,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Calculus
High School Interest

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