Income from Winning the LotteryDate: 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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