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To solve this problem, we’ll use the compound interest formula:

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

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

### Given Values:
- $ P = 45000 $
- $ r = 0.10 $ (10% interest rate)
- $ t = 5 $ years

We need to calculate the final amount for four different compounding methods: annual, semiannual, monthly, and daily. 

Let’s go through each case:

1. **Annual Compounding** ($ n = 1 $):
   \[
   A = 45000 \left(1 + \frac{0.10}{1}\right)^{1 \times 5} = 45000 \times (1.10)^5
   \]

2. **Semiannual Compounding** ($ n = 2 $):
   \[
   A = 45000 \left(1 + \frac{0.10}{2}\right)^{2 \times 5} = 45000 \times \left(1.05\right)^{10}
   \]

3. **Monthly Compounding** ($ n = 12 $):
   \[
   A = 45000 \left(1 + \frac{0.10}{12}\right)^{12 \times 5} = 45000 \times \left(1 + \frac{0.10}{12}\right)^{60}
   \]

4. **Daily Compounding** ($ n = 365 $):
   \[
   A = 45000 \left(1 + \frac{0.10}{365}\right)^{365 \times 5} = 45000 \times \left(1 + \frac{0.10}{365}\right)^{1825}
   \]

I'll calculate each value to the nearest cent.

#### Calculations:
Let's compute these values.Here are the investment values at the end of 5 years, rounded to the nearest cent, for each compounding method:

1. **Annual Compounding**: $72,472.95
2. **Semiannual Compounding**: $73,300.26
3. **Monthly Compounding**: $74,038.90
4. **Daily Compounding**: $74,187.38

Would you like further details on the calculations or have any questions?

### Related Questions
1. What would be the investment value if compounded quarterly?
2. How would a lower interest rate, like 5%, affect these results?
3. How does the duration of the investment (e.g., 10 years instead of 5) impact the results for each compounding method?
4. What if the principal was $100,000 instead of $45,000?
5. How do continuous compounding calculations compare to these results?

### Tip
The more frequently interest is compounded, the more interest will be earned over the same period, due to the effect of compounding on compounding.

If 45000 dollars is invested at an interest rate of 10 percent per year, find the value of the investment at the end of 5 years for the following compounding methods, to the nearest cent.

Calculate the future value of a $45,000 investment at a 10% annual interest rate over 5 years with various compounding methods: annually, semiannually, monthly, and daily. Learn the compound interest formula and see how different compounding frequencies affect the final investment value.

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