Find the compound amount for the certificate of deposit. Assume daily compounding using the following table. Round to the nearest cent.



Amount: $3,112
Rate: 4%
Years: 5
Select one:

A.
$4,642.46

B.
$3,800.96

C.
$3,800.95

D.
$3,995.82

To find the compound amount with daily compounding, we use the formula:

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

Where:
- $ A $ is the amount of money accumulated after n years, including interest.
- $ P $ is the principal amount (the initial amount of money).
- $ 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.

Given:
- Principal amount ($ P $) = $3,112
- Annual interest rate ($ r $) = 4% = 0.04
- Time ($ t $) = 5 years
- Number of times interest is compounded per year ($ n $) = 365 (since it’s daily compounding)

Plugging these values into the formula, we get:

\[
A = 3112 \left(1 + \frac{0.04}{365}\right)^{365 \times 5}
\]

First, calculate the daily interest rate:

\[
\frac{0.04}{365} \approx 0.000109589
\]

Next, calculate the exponent:

\[
365 \times 5 = 1825
\]

Now, compute the compound amount:

\[
A = 3112 \left(1 + 0.000109589\right)^{1825}
\]

First, calculate the term inside the parentheses:

\[
1 + 0.000109589 \approx 1.000109589
\]

Raise this term to the 1825th power:

\[
1.000109589^{1825} \approx 1.221386025
\]

Finally, multiply this result by the principal amount:

\[
A = 3112 \times 1.221386025 \approx 3800.96
\]

Rounding to the nearest cent, the compound amount is approximately $3,800.96.

Thus, the correct answer is:

**B. $3,800.96**

Would you like more details or have any questions about this calculation?

Here are some related questions:
1. What is the effect of compounding frequency on the compound amount?
2. How would the compound amount change if the rate were higher?
3. What is the formula for compound interest with monthly compounding?
4. How does the length of the investment period affect the compound amount?
5. Can the compound amount ever be less than the principal?
6. What are some practical uses of compound interest in personal finance?
7. How does compound interest differ from simple interest?
8. What would the compound amount be for different principal amounts but the same rate and time period?

**Tip:** Compounding more frequently results in more interest accrued over time. Daily compounding typically yields a higher amount than monthly or annual compounding.

Learn how to calculate compound interest with daily compounding using a detailed example problem. Understand the formula A = P (1 + r/n)^(nt) with practical application. Suitable for high school students and anyone interested in financial mathematics.

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