Melanie places $6,000 into an account that earns 4% interest compounded daily. Approximately six months later (180 days), she deposits an additional $3,000 in the same account. What is the value of the account after 1 year? (Assume all years have 365 days. Round your answer to the nearest cent.)

To find the value of Melanie's account after 1 year, we need to account for the interest earned on both deposits. Here's the step-by-step solution:

### Step 1: Calculate the value of the first deposit after 180 days
- **Initial deposit:** $6,000
- **Interest rate:** 4% annually, compounded daily
- **Number of days for the first deposit:** 180 days

The formula for compound interest is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the amount of money accumulated after $ n $ years, including interest.
- $ P $ is the principal amount (the initial deposit).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested for (in years).

For the first deposit after 180 days:

\[
A_1 = 6000 \left(1 + \frac{0.04}{365}\right)^{180}
\]

### Step 2: Calculate the value of the first deposit after 365 days
After 180 days, Melanie deposits an additional $3,000. So, we calculate the new balance after 365 days.

First, calculate the amount from Step 1 to continue compounding for the remaining 185 days:

\[
A_1^{\prime} = A_1 \left(1 + \frac{0.04}{365}\right)^{185}
\]

### Step 3: Calculate the value of the second deposit after 185 days
- **Second deposit:** $3,000
- **Number of days for the second deposit:** 185 days

The amount after 185 days for the second deposit:

\[
A_2 = 3000 \left(1 + \frac{0.04}{365}\right)^{185}
\]

### Step 4: Add the amounts from both deposits
The total amount in the account after 1 year:

\[
A_{\text{total}} = A_1^{\prime} + A_2
\]

Now, let's calculate these values step by step.

### Step 1: Calculate the value of the first deposit after 180 days
\[
A_1 = 6000 \left(1 + \frac{0.04}{365}\right)^{180} \approx 6000 \times 1.019945 = 6119.67
\]

### Step 2: Calculate the value of the first deposit after 365 days
\[
A_1^{\prime} = 6119.67 \left(1 + \frac{0.04}{365}\right)^{185} \approx 6119.67 \times 1.020319 = 6243.81
\]

### Step 3: Calculate the value of the second deposit after 185 days
\[
A_2 = 3000 \left(1 + \frac{0.04}{365}\right)^{185} \approx 3000 \times 1.020319 = 3060.96
\]

### Step 4: Add the amounts from both deposits
\[
A_{\text{total}} = 6243.81 + 3060.96 = 9304.77
\]

So, the value of the account after 1 year is **$9,304.77**.

Would you like to ask for more details or explanations?

Here are 5 related questions you might find interesting:
1. How would the total amount change if the interest were compounded monthly instead of daily?
2. What would be the value of the account if the second deposit was made after 1 year instead of 6 months?
3. How would the value change if the interest rate were increased to 5%?
4. Can you find the future value if Melanie decided to withdraw $1,000 after 9 months?
5. What would happen if no additional deposit were made after 6 months?

**Tip:** When dealing with compound interest, remember to adjust the time and rate according to how often the interest is compounded.

Discover how to calculate the total value of an account after 1 year with compound interest. Melanie deposits $6,000 initially and $3,000 after 6 months into an account earning 4% interest compounded daily. Learn the step-by-step solution and explore variations in interest rates and deposit timings.

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