A student has saved 50000 in the bank for her college tuition. When she starts c

Question

A student has saved 50000 in the bank for her college tuition. When she starts college, she invests money in a savings account that pays 1.4 percent per year, compounded continuously. Suppose her college tuition is 22500 per year and she arranges with the college that the money will be deducted from her savings account continuously, but they will drop her from all her classes the moment the money runs out. We wish to find out how long she can stay in school before she runs out of money. Let y(t) represents the amount of money in her account at time t (t is in years). y(0) = _______ (dollars) With y representing the amount of money in dollars at time t (in years) write a differential equation which models this situation. y' = f(t, y) = _______. Find an equation for the amount of money in the account at time t, where t is the number of years since starting college. y(t) = _______ Find the number of years before the account is empty: t = _______

Explanation / Answer

At y(0) the money will be $50000 only as she begins with this money originally

As money is increasing and decreasing continuosuly

y(t) = (50000)e^1.4t - 22500e^t

y'(t)= 1.4*50000e^1.4t - 1.4*22500e^t

y' = 70000e^1.4t - 31500e^t = 70000e^te^{0.4t} - 31500e^t

y' = 3500e^t(20e^0.4t-9)

Also

y(t)= (50000)e^1.4t - 22500e^t

When the account is empty

(50000)e^1.4t - 22500e^t = 0

Add 22500e^t to both sides

50000e^{1.4t}-22500e^t+22500e^t=0+22500e^t

50000e^{1.4t} = 22500e^t

ln(50000e^{1.4t})=ln(22500e^t)

t = [20ln(3) - ln(1.024E13]/4

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