We describe a savings account with continuous withdrawals that earns interest compounded continuously. Let denote the money in the account, years after it was open. Find the differer
equation and initial condition that satisfies.
Initial amount , annual interest rate , withdrawals at the rate of per month

Ask by Evans Simmons. in the United States

Mar 23,2025

Question

We describe a savings account with continuous withdrawals that earns interest compounded continuously. Let denote the money in the account, years after it was open. Find the differer
equation and initial condition that satisfies.
Initial amount , annual interest rate , withdrawals at the rate of per month

Ask by Evans Simmons. in the United States

Mar 23,2025

UpStudy AI · Accepted Answer

We are given an account with an initial amount of $$ \$13,\!000 $$, earning interest at an annual rate of $$ r = 0.02 $$ (compounded continuously), and with continuous withdrawals at the rate of $$ \$500 $$ per month. Since withdrawals are monthly, we convert this rate to an annual rate:

$$
\$500 \text{ per month} = 12 \times \$500 = \$6000 \text{ per year.}
$$

Let $$ y = P(t) $$ denote the balance in the account as a function of time $$ t $$ (in years). The changes in $$ P(t) $$ are due to the interest earned and the withdrawals made. The differential equation is set up as follows:

1. The interest contributes a rate of change given by $$ 0.02 P(t) $$ (because the interest is 2% per year).
2. The withdrawals contribute a negative rate of change of $$ \$6000 $$ per year.

Thus, the differential equation that $$ P(t) $$ satisfies is

$$
\frac{dP}{dt} = 0.02P - 6000.
$$

The initial condition is given by the initial deposit:

$$
P(0) = 13000.
$$
The differential equation for the account balance $$ P(t) $$ is: $$ \frac{dP}{dt} = 0.02P - 6000 $$ with the initial condition: $$ P(0) = 13000 $$

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