To find the amount of interest Kaira will earn at the end of the 3-year term, we first need to calculate the total value of the investment after 3 years using the provided formula \( C(x) = 600x^3 + 600x^2 + 600x \).
Here, \( x \) is defined as \( 1 + r \), where \( r \) is the interest rate. Given that the interest rate \( r \) is 4.75%, we can express \( x \) as:
\[ x = 1 + 0.0475 = 1.0475 \]
Now we need to evaluate \( C(x) \):
\[ C(1.0475) = 600(1.0475^3) + 600(1.0475^2) + 600(1.0475) \]
Calculating each term:
-
Calculate \( 1.0475^3 \): \[ 1.0475^3 \approx 1.148882 \]
-
Calculate \( 1.0475^2 \): \[ 1.0475^2 \approx 1.097616 \]
-
Calculate \( 1.0475 \): \[ 1.0475 \approx 1.0475 \]
Now put these values into the equation:
\[ C(1.0475) \approx 600(1.148882) + 600(1.097616) + 600(1.0475) \]
Calculating each term:
- \( 600 \times 1.148882 \approx 689.3292 \)
- \( 600 \times 1.097616 \approx 658.5696 \)
- \( 600 \times 1.0475 \approx 628.5 \)
Now sum these amounts:
\[ C(1.0475) \approx 689.3292 + 658.5696 + 628.5 \approx 1976.3988 \]
So the total value of the investment at the end of 3 years is approximately \( 1976.40 \).
Next, calculate the total deposits:
Total deposits = 3 years * $600 = $1800.
Now, we find the interest earned:
\[ \text{Interest} = C(1.0475) - \text{Total deposits} = 1976.40 - 1800 = 176.40 \]
Thus, rounding to two decimal places, the amount of interest that Kaira will earn at the end of the 3-year term is
\[ \boxed{176.40} \]