Suppose that money is deposited daily into a savings account at an annual rate of $800. If the account pays 4% interest compounded continuously, estimate the balance in the account at the end of 3 years.

To estimate the balance in the account at the end of 3 years, we can model this situation using a differential equation that represents continuous compounding with continuous contributions.

Given:
- Daily deposit rate: $\$800 \text{ per year}$
- Interest rate: $r = 4\% = 0.04$
- Time period: $t = 3$ years

The differential equation that models the balance $B(t)$ in the account over time is:

\[
\frac{dB(t)}{dt} = 800 + 0.04B(t)
\]

This equation states that the rate of change of the balance is equal to the sum of the daily deposits and the interest accumulated on the current balance.

### Step 1: Solve the differential equation
To solve this, we first need to find the integrating factor, $ \mu(t) $:

\[
\mu(t) = e^{\int 0.04 \, dt} = e^{0.04t}
\]

Multiplying both sides of the differential equation by the integrating factor:

\[
e^{0.04t} \frac{dB(t)}{dt} = 800e^{0.04t} + 0.04B(t) e^{0.04t}
\]

The left side can now be expressed as the derivative of the product:

\[
\frac{d}{dt}\left(B(t)e^{0.04t}\right) = 800e^{0.04t}
\]

### Step 2: Integrate both sides

\[
\int \frac{d}{dt}\left(B(t)e^{0.04t}\right) \, dt = \int 800e^{0.04t} \, dt
\]

\[
B(t)e^{0.04t} = \frac{800}{0.04} e^{0.04t} + C
\]

\[
B(t)e^{0.04t} = 20000e^{0.04t} + C
\]

### Step 3: Solve for $ B(t) $

\[
B(t) = 20000 + Ce^{-0.04t}
\]

To determine the constant $C$, we use the initial condition. Assuming the initial balance $B(0) = 0$:

\[
B(0) = 20000 + C = 0 \implies C = -20000
\]

Thus, the balance function is:

\[
B(t) = 20000(1 - e^{-0.04t})
\]

### Step 4: Calculate the balance at $ t = 3 $ years

\[
B(3) = 20000(1 - e^{-0.12}) 
\]

Now, evaluate this expression:

\[
B(3) \approx 20000(1 - e^{-0.12}) \approx 20000(1 - 0.88692) \approx 20000 \times 0.11308 \approx 2261.60
\]

So, the balance in the account at the end of 3 years is approximately **$2,261.60**.

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Would you like further details on this process, or do you have any questions?

### Relative Questions:
1. How does the continuous compounding formula differ from the discrete one?
2. What would be the balance after 5 years with the same conditions?
3. How would the balance change if the interest rate were 5% instead of 4%?
4. Can you derive the differential equation for continuous deposits with different interest rates?
5. How does the deposit frequency affect the final balance in a continuously compounded interest scenario?

### Tip:
When dealing with continuously compounded interest, understanding the differential equation that models the scenario can greatly simplify the calculation of future balances.

Learn how to estimate the balance in a savings account with daily deposits and continuous compounding interest at 4% over 3 years. This problem involves solving a differential equation and applying concepts of continuous compounding. Ideal for undergraduate students.

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