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8. [tex]$\$[/tex]650[tex]$ is invested for two years at a rate of $[/tex]8\%[tex]$ per year, compounding annually.

8a. Complete the table below to determine the final value of the investment, rounding your answers to the nearest cent.

| Year | Principal | Interest Earned |
|-------------|---------------|-------------------|
| First year | $[/tex]\[tex]$650$[/tex] | [tex]$\square$[/tex] |
| Second year | [tex]$\$[/tex]650 + \[tex]$\square$[/tex] | [tex]$\square$[/tex] |
| Third year | [tex]$\$[/tex]\square + \[tex]$56.16$[/tex] | [tex]$\square$[/tex] |


Sagot :

Certainly! Let's go through this step-by-step to determine the final value of the investment after two years, with an annual compounding interest rate of 8%, and round our answers to the nearest cent.

### Initial Investment
- Principal: [tex]\( \$ 650 \)[/tex]

### First Year
1. Calculate the Interest Earned in the First Year:
- Interest Rate: 8%
- Interest Earned = Principal [tex]\(\times\)[/tex] Interest Rate
[tex]\[ \text{Interest Earned} = 650 \times 0.08 = 52.00 \][/tex]

2. Calculate the Total at the End of the First Year:
- Total = Principal [tex]\(\+\)[/tex] Interest Earned
[tex]\[ \text{First Year Total} = 650 + 52.00 = 702.00 \][/tex]

### Second Year
1. Calculate the Interest Earned in the Second Year:
- New Principal = First Year Total = [tex]\( \$ 702 \)[/tex]
- Interest Earned = New Principal [tex]\(\times\)[/tex] Interest Rate
[tex]\[ \text{Interest Earned} = 702.00 \times 0.08 = 56.16 \][/tex]

2. Calculate the Total at the End of the Second Year:
- Total = New Principal [tex]\(\+\)[/tex] Interest Earned
[tex]\[ \text{Second Year Total} = 702.00 + 56.16 = 758.16 \][/tex]

### Summary
Let's populate our table now:

| Year | Principal | Interest Earned | Total |
|--------------|-------------|-----------------|-------------|
| First Year | \[tex]$650 | \$[/tex]52.00 | \[tex]$702.00 | | Second Year | \$[/tex]702.00 | \[tex]$56.16 | \$[/tex]758.16 |

In more detail:

1. First Year:
- Principal: [tex]\( \$ 650 \)[/tex]
- Interest Earned: [tex]\( \$ 52.00 \)[/tex]
- Total: [tex]\( \$ 702.00 \)[/tex]

2. Second Year:
- Principal: [tex]\( \$ 702.00 \)[/tex] (which is the total from the end of the first year)
- Interest Earned: [tex]\( \$ 56.16 \)[/tex]
- Total: [tex]\( \$ 758.16 \)[/tex]

### Filling in the Table
Given the question prompts:
- First Year:
[tex]\[ \text{Principal}: \$ 650 \text{, Interest Earned: } \$ 52.00 \text{, Total: } \$ 702.00 \][/tex]

- Second Year:
- [tex]$ \text{Principal at start of the second year: } \$[/tex] 702.00 [tex]$ \[ \text{Interest Earned: } \$[/tex] 56.16
\]
[tex]\[ \text{Total at the end of the second year: } \$ 758.16 \][/tex]

Thus, the completed table will be:

| Year | Principal | Interest Earned | Total |
|--------------|-------------|------------------|--------------|
| First Year | \[tex]$650 | \$[/tex]52.00 | \[tex]$702.00 | | Second Year | \$[/tex]702.00 | \[tex]$56.16 | \$[/tex]758.16 |

And the final value of the investment after two years is: [tex]\( \$ 758.16 \)[/tex].