Question

An investor transfers $400,000 into an IRA at age 60. The account pays 3.25% interest compounded continuously. He plans to withdraw $24,000 each year from the account in a near-continuous manner until the account is depleted. (1) Write a differential equation to describe the situation. (2) How much will be left in the account after 10 years? (3) When will the account be completely depleted?

Answer 1

The differential equation for the situation is dy/dt = r * y - w, where r is the interest rate, y is the amount of money in the account, and w is the withdrawal amount. To find out how much will be left in the account after 10 years, solve the differential equation using the given values. To determine when the account will be completely depleted, continue solving the differential equation until the value of y becomes zero.

Step-by-step explanation:

(1) The differential equation for the situation can be expressed as:

dy/dt = r * y - w

where:

• dy/dt represents the rate of change of y (the amount of money in the account) with respect to time t.
• r represents the interest rate (3.25%).
• y represents the amount of money in the account.
• w represents the withdrawal amount ($24,000/year).

(2) To find out how much will be left in the account after 10 years, we can solve the differential equation using an initial condition of y(0) = $400,000 and the given values of r and w. The solution will give us the value of y after 10 years.

(3) To determine when the account will be completely depleted, we can continue solving the differential equation until the value of y becomes zero (or negative), which indicates that the account is empty.