Your crazy , rich uncle (who has studied differential equations and loves math)

Question

Your crazy , rich uncle (who has studied differential equations and loves math) opens a new bank that compounds interest continuously . He decides to set up a college fund for all of his nieces and nephews. In your account, which pays interest at a 5% annual rate, he deposits $10,000. He also adds money continuously to the account , at a rate of $50 per month.

a. find a formula for the amount of money in the account "t" years after the initial $10,000 deposit.

b. How long will it take until the account balance is $100,000?

c 1. Of course you're already in college, so you arrange for $100 per month to be transferred (continuously of course) from this fund to your checking account to pay for various college expense, find a formula for the amount of money in the account "t" years after the initial $10,000 deposit.

c 2. when will the account balance reach $10?

Explanation / Answer

a) We will use t in years. So she deposits 50 a month which is 12*50 = 600 a year. So, if we let Y be the balance of the account, the differential equation would look like this: dY/dt = .05Y + 600 This equation is separable, so we solve it like this: dY/(Y+12000) = .05dt Integrating both sides gives: ln(Y+12000) = .05t + c Solving for y gives: Y = Ke^(.05t) - 12000 Knowing that Y(0) = 10000 allows us to solve for K and get: Y = 22000e^(.05t) - 12000 b) To find t we simply plug in 100000 for Y and solve for t: 100000 = 22000e^(.05t) - 12000 112000 = 22000e^(.05t) 56/11 = e^(.05t) ln(56/11) = .05t t = ln(56/11)/.05 which is about 32.549 years. c) Now, with the 100 withdrawal every month, added to the 50 deposit each month, she will be losing 600 dollars every year. So the differential form will look like this: dY/dt = .05Y - 600 We solve this just like the first one to get: Y = -2000e^(.05t) + 12000 d) Plug in 10 for Y and solve for t: 10 = -2000e^(.05t) + 12000 -11990 = -2000e^(.05t) 1199/200 = e^(.05t) ln(1199/200) = .05t t = ln(1199/200)/.05 which is about 35.819 years.

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