https://file.aiquickdraw.com/mathbot/file/17294387112544cs66qbb.jpg

The problem you're dealing with involves compound interest. The formula for compound interest is:

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

Where:
- $A$ is the amount of money in the account after $t$ years.
- $P$ is the principal amount (initial investment).
- $r$ is the annual interest rate (in decimal form).
- $n$ is the number of times the interest is compounded per year.
- $t$ is the number of years.

From the image, you are given:
- Principal $P = 5000$
- Annual interest rate $r = 5\% = 0.05$
- The interest is compounded semiannually, so $n = 2$
- Time $t = 4$ years

### Step 1: Calculate $A$
Now we substitute the values into the formula:

\[
A = 5000 \left(1 + \frac{0.05}{2}\right)^{2 \times 4}
\]
\[
A = 5000 \left(1 + 0.025\right)^8 = 5000 \left(1.025\right)^8
\]
\[
A = 5000 \times 1.2184 = 6092.02
\]

So, the amount in the account after 4 years is approximately **\$6092.02**.

### Step 2: Calculate the interest earned
The interest earned is the total amount minus the principal:

\[
\text{Interest earned} = A - P = 6092.02 - 5000 = 1092.02
\]

So, the interest earned over 4 years is **\$1092.02**.

Would you like a detailed breakdown of the steps? Let me know if you'd like to go over any specific part.

### Related Questions:
1. How does increasing the frequency of compounding affect the total amount?
2. What would the balance be if the interest was compounded quarterly?
3. How would the total amount change if the interest rate were 6% instead of 5%?
4. What if the time period was extended to 6 years?
5. How much interest would be earned if the account had a principal of \$10,000 instead?

**Tip:** Always double-check whether the rate provided is already in decimal form or needs to be converted from a percentage before plugging it into the formula.

Find how much money will be in the account after the given number of years. The principal is $5000, the rate is 5%, interest is compounded semiannually, and the time is 4 years.

Calculate how much money will be in the account after 4 years with a principal of $5000 at 5% interest, compounded semiannually. This problem demonstrates the compound interest formula and is suitable for students in Grades 9-12.

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