Question
Modeling with Polynomials Practice
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Question
For the past 3 years, Kaira has deposited $600 at the beginning of each year into an investment account with an interest rate of 4.75%. Use x=1+r
, where r is the interest rate, and the equation C(x)=600x3+600x2+600x
. The equation represents the relationship between C(x)
, the value of the investment after 3 years. Given that the amount of interest earned is the difference between the total value of the investment after 3 years and the sum of her $600 depositsl, find the amount of interest that Kaira will earn at the end of the 3-year term. Round the answer to two decimal places.(1 point)
The amount of interest that Kaira will earn at the end of the 3-year term is $
.
Answers
GPT-4o mini
To find the amount of interest that Kaira will earn at the end of the 3-year term, we first need to evaluate the expression for \( C(x) \) given \( x = 1 + r \), where \( r \) is the interest rate (4.75% or 0.0475).
Let's calculate \( x \):
\[
x = 1 + r = 1 + 0.0475 = 1.0475
\]
Next, we will substitute \( x \) into the equation:
\[
C(x) = 600x^3 + 600x^2 + 600x
\]
Now, we'll calculate \( C(1.0475) \):
1. **Calculate \( x^3 \)**:
\[
x^3 = (1.0475)^3 \approx 1.148882
\]
2. **Calculate \( x^2 \)**:
\[
x^2 = (1.0475)^2 \approx 1.096829
\]
3. **Calculating \( C(1.0475) \)**:
\[
C(1.0475) = 600(1.148882) + 600(1.096829) + 600(1.0475)
\]
Now we compute each term:
- \( 600 \times 1.148882 \approx 689.3292 \)
- \( 600 \times 1.096829 \approx 658.0974 \)
- \( 600 \times 1.0475 \approx 628.5 \)
Next, we sum these values:
\[
C(1.0475) \approx 689.3292 + 658.0974 + 628.5 \approx 1975.9266
\]
So, the total value of the investment after 3 years is approximately \$1975.93.
Next, we calculate the total amount Kaira deposited over the 3 years:
\[
\text{Total Deposits} = 3 \times 600 = 1800
\]
Finally, the interest earned is the difference between the total value of the investment and the total deposits:
\[
\text{Interest Earned} = C(1.0475) - \text{Total Deposits}
\]
\[
\text{Interest Earned} \approx 1975.93 - 1800 = 175.93
\]
Thus, the amount of interest that Kaira will earn at the end of the 3-year term is approximately **$175.93**.
Let's calculate \( x \):
\[
x = 1 + r = 1 + 0.0475 = 1.0475
\]
Next, we will substitute \( x \) into the equation:
\[
C(x) = 600x^3 + 600x^2 + 600x
\]
Now, we'll calculate \( C(1.0475) \):
1. **Calculate \( x^3 \)**:
\[
x^3 = (1.0475)^3 \approx 1.148882
\]
2. **Calculate \( x^2 \)**:
\[
x^2 = (1.0475)^2 \approx 1.096829
\]
3. **Calculating \( C(1.0475) \)**:
\[
C(1.0475) = 600(1.148882) + 600(1.096829) + 600(1.0475)
\]
Now we compute each term:
- \( 600 \times 1.148882 \approx 689.3292 \)
- \( 600 \times 1.096829 \approx 658.0974 \)
- \( 600 \times 1.0475 \approx 628.5 \)
Next, we sum these values:
\[
C(1.0475) \approx 689.3292 + 658.0974 + 628.5 \approx 1975.9266
\]
So, the total value of the investment after 3 years is approximately \$1975.93.
Next, we calculate the total amount Kaira deposited over the 3 years:
\[
\text{Total Deposits} = 3 \times 600 = 1800
\]
Finally, the interest earned is the difference between the total value of the investment and the total deposits:
\[
\text{Interest Earned} = C(1.0475) - \text{Total Deposits}
\]
\[
\text{Interest Earned} \approx 1975.93 - 1800 = 175.93
\]
Thus, the amount of interest that Kaira will earn at the end of the 3-year term is approximately **$175.93**.