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

Question

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

Answer 1

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 2

Are you sure

Answer 3

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 4

What is the correct answer

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!