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
\[C(x)=600 x^3+600 x^2+600 x\]

Answer 1

Are you sure

Answer 2

What is the correct answer

Answer 3

To interpret the given problem, we need to understand what is happening with Kaira's deposits and how the interest is applied. Kaira deposits \$600 at the beginning of each year for 3 years, and the interest rate is 4.75%.

Using \( x = 1 + r \), where \( r \) is the interest rate, we can calculate:

\[
r = 0.0475 \quad \text{(4.75% expressed in decimal)}
\]
\[
x = 1 + r = 1 + 0.0475 = 1.0475
\]

Now, let's look at the equation given:

\[
C(x) = 600 x^3 + 600 x^2 + 600 x
\]

This equation represents the future value of Kaira's investments after 3 years, accounting for the interest earned on each deposit:

- \( 600 x^3 \) represents the future value of the deposit made at the beginning of year 1 after 3 years.
- \( 600 x^2 \) represents the future value of the deposit made at the beginning of year 2 after 2 years.
- \( 600 x \) represents the future value of the deposit made at the beginning of year 3 after 1 year.

Now we can calculate \( C(x) \):

1. Substitute \( x \) into the equation:
\[
C(1.0475) = 600 (1.0475^3) + 600 (1.0475^2) + 600 (1.0475)
\]

2. Calculate each term:

- \( 1.0475^3 \):
\[
1.0475^3 \approx 1.148882
\]
- \( 1.0475^2 \):
\[
1.0475^2 \approx 1.097707
\]
- \( 1.0475 \):
\[
1.0475 \approx 1.0475
\]

3. Now substitute back into the equation:
\[
C(1.0475) = 600 \times 1.148882 + 600 \times 1.097707 + 600 \times 1.0475
\]
- Compute each multiplication:
\[
600 \times 1.148882 \approx 689.3292
\]
\[
600 \times 1.097707 \approx 658.622
\]
\[
600 \times 1.0475 \approx 628.5
\]

4. Finally, sum these values:
\[
C(1.0475) \approx 689.3292 + 658.622 + 628.5 \approx 1976.4512
\]

Thus, the total value of Kaira's investment account after 3 years is approximately:

\[
\boxed{1976.45}
\]

Answer 4

Let's go through the calculations again step by step to ensure accuracy. Kaira deposits \$600 at the beginning of each of the 3 years into an investment account with an interest rate of 4.75%.

Using the formula:

\[
C(x) = 600 x^3 + 600 x^2 + 600 x
\]

Where \( x = 1 + r \) and \( r = 0.0475 \):

\[
x = 1 + 0.0475 = 1.0475
\]

Now let's calculate \( C(1.0475) \):

1. **Calculate each term separately.**

- Calculate \( x^3 \):
\[
1.0475^3 \approx 1.148882
\]

- Calculate \( x^2 \):
\[
1.0475^2 \approx 1.097507
\]

- Calculate \( x \):
\[
1.0475 = 1.0475
\]

2. **Calculate the contributions of each deposit into the future value:**

- For the first deposit (after 3 years):
\[
600 \times 1.148882 \approx 689.3292
\]

- For the second deposit (after 2 years):
\[
600 \times 1.097507 \approx 658.5042
\]

- For the third deposit (after 1 year):
\[
600 \times 1.0475 \approx 628.5
\]

3. **Summing these values:**
\[
C(1.0475) = 689.3292 + 658.5042 + 628.5
\]

Calculate the total:
\[
C(1.0475) \approx 689.3292 + 658.5042 + 628.5 \approx 1976.3336
\]

So, the corrected total value of Kaira's investment account after 3 years is approximately:

\[
\boxed{1976.33}
\]

Thank you for prompting a review of the calculation, and I apologize for any confusion in the previous response.

Answer 5

Let's carefully re-evaluate the calculation step-by-step to find the correct total amount in Kaira's investment account after 3 years.

### Information Given
- **Annual Deposit**: $600
- **Interest Rate**: 4.75% (or 0.0475 as a decimal)
- **Formula Given**:
\[
C(x) = 600 x^3 + 600 x^2 + 600 x
\]
- **Substituting \( x = 1 + r \)**:
\[
x = 1 + 0.0475 = 1.0475
\]

### Calculation Steps

1. **Calculate \( x^3 \), \( x^2 \), and \( x \)**:
- \( x^3 = (1.0475)^3 \)
- \( x^2 = (1.0475)^2 \)
- \( x = 1.0475 \)

Now performing these calculations:

- \( x^3 \):
\[
(1.0475)^3 \approx 1.148882
\]
- \( x^2 \):
\[
(1.0475)^2 \approx 1.097756625
\]
- \( x \):
\[
(1.0475) \approx 1.0475
\]

2. **Compute Future Value for Each Deposit**:
- Contribution of the deposit at the beginning of year 1 (after 3 years):
\[
600 \times 1.148882 \approx 689.3292
\]

- Contribution of the deposit at the beginning of year 2 (after 2 years):
\[
600 \times 1.097756625 \approx 658.653975
\]

- Contribution of the deposit at the beginning of year 3 (after 1 year):
\[
600 \times 1.0475 \approx 628.5
\]

3. **Total Future Value**:
Now sum these future values:
\[
C(1.0475) = 689.3292 + 658.653975 + 628.5
\]
- Performing the addition:
\[
C(1.0475) \approx 689.3292 + 658.653975 + 628.5 \approx 1976.483175
\]

4. **Final Answer**:
The total value of Kaira's investment account after 3 years is approximately:

\[
\boxed{1976.48}
\]

Thank you for your patience, and this should be the correct calculation!