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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 $

5 answers

To calculate the amount of interest that Kaira will earn at the end of the 3-year term, we'll first consider the deposits she made and the interest earned on each:

  1. Kaira deposits $600 at the beginning of each year for 3 years.
  2. The interest rate is 4.75%, which we will convert to a decimal \( r = 0.0475 \).
  3. Using the formula \( x = 1 + r = 1 + 0.0475 = 1.0475 \).

Now we plug this into the equation provided for \( C(x) \): \[ C(x) = 600 x^3 + 600 x^2 + 600 x \]

Calculating each term:

  • For the first deposit (made at the beginning of Year 1), it earns interest for 3 years: \[ 600 x^3 = 600 (1.0475)^3 \]
  • For the second deposit (made at the beginning of Year 2), it earns interest for 2 years: \[ 600 x^2 = 600 (1.0475)^2 \]
  • For the third deposit (made at the beginning of Year 3), it earns interest for 1 year: \[ 600 x = 600 (1.0475) \]

Now, let's calculate each component:

Calculate \( (1.0475)^3 \):

\[ (1.0475)^3 \approx 1.148882 \]

So, \[ 600 \cdot 1.148882 \approx 689.33 \]

Calculate \( (1.0475)^2 \):

\[ (1.0475)^2 \approx 1.09796 \]

So, \[ 600 \cdot 1.09796 \approx 658.78 \]

Calculate \( (1.0475) \):

So, \[ 600 \cdot 1.0475 \approx 628.50 \]

Now, combine these values to get \( C(x) \):

\[ C(x) \approx 689.33 + 658.78 + 628.50 = 1976.61 \]

Calculate the total deposits:

Kaira deposited a total of 3 deposits of $600 each: \[ \text{Total Deposits} = 3 \cdot 600 = 1800 \]

Calculate the interest earned:

\[ \text{Interest Earned} = C(x) - \text{Total Deposits} = 1976.61 - 1800 = 176.61 \]

Conclusion:

The amount of interest that Kaira will earn at the end of the 3-year term is approximately $176.61 (rounded to two decimal places).

That is wrong

Let’s re-evaluate the calculations step by step to determine the correct interest earned by Kaira over the 3-year term.

Given:

  • Annual deposit: \( P = 600 \)
  • Interest rate: \( r = 4.75% = 0.0475 \)
  • Number of years: \( t = 3 \)

Formula:

The formula for the total future value \( C(x) \) after 3 years when making deposits at the beginning of the year, using \( x = 1 + r \) is: \[ C(x) = 600x^3 + 600x^2 + 600x \] Where \( x = 1.0475 \).

Calculating each term:

  1. Calculate \( x^3 \): \[ x^3 = (1.0475)^3 \approx 1.148882 \] \[ 600x^3 = 600 \times 1.148882 \approx 689.33 \]

  2. Calculate \( x^2 \): \[ x^2 = (1.0475)^2 \approx 1.09756 \] \[ 600x^2 = 600 \times 1.09756 \approx 658.54 \]

  3. Calculate \( x \): \[ x = 1.0475 \] \[ 600x = 600 \times 1.0475 \approx 628.50 \]

Total Value After 3 Years:

Now, summing all the amounts earned from each deposit: \[ C(x) \approx 689.33 + 658.54 + 628.50 \approx 1976.37 \]

Total Deposits:

Total amount deposited over 3 years: \[ \text{Total Deposits} = 3 \times 600 = 1800 \]

Interest Earned:

The interest earned will be the total value minus the total deposits: \[ \text{Interest} = C(x) - \text{Total Deposits} \] \[ = 1976.37 - 1800 = 176.37 \]

Conclusion:

The amount of interest that Kaira will earn at the end of the 3-year term is approximately $176.37 (rounded to two decimal places).

still wrong

Let's carefully recalculate the future value of Kaira's investment, step by step, and ensure every detail is accurately addressed.

Given Information

  • Kaira deposits \( P = 600 \) at the beginning of each year.
  • Interest rate \( r = 4.75% \) or \( r = 0.0475 \).
  • Number of years \( t = 3 \).

The formula \( C(x) = 600 x^3 + 600 x^2 + 600 x \), where \( x = 1 + r = 1.0475 \), represents the future value of the investments after 3 years.

Calculations

Step 1: Compute \( x \)

\[ x = 1 + r = 1 + 0.0475 = 1.0475 \]

Step 2: Compute each term in \( C(x) \)

  1. Calculate \( x^3 \): \[ x^3 = (1.0475)^3 \approx 1.148882 \] \[ 600 x^3 = 600 \times 1.148882 \approx 689.33 \]

  2. Calculate \( x^2 \): \[ x^2 = (1.0475)^2 \approx 1.09756 \] \[ 600 x^2 = 600 \times 1.09756 \approx 658.54 \]

  3. Calculate \( x^1 \): \[ x^1 = (1.0475) \approx 1.0475 \] \[ 600 x = 600 \times 1.0475 \approx 628.50 \]

Step 3: Calculate Total Future Value \( C(x) \)

Now we add these values together: \[ C(x) \approx 689.33 + 658.54 + 628.50 \approx 1976.37 \]

Step 4: Calculate Total Amount Deposited \[ \text{Total Deposits} = 3 \times 600 = 1800 \]

Step 5: Calculate Total Interest Earned Now we find the interest earned: \[ \text{Interest Earned} = C(x) - \text{Total Deposits} \] \[ \text{Interest Earned} \approx 1976.37 - 1800 = 176.37 \]

Final Answer

The total interest that Kaira will earn at the end of the 3-year term is $176.37 (rounded to two decimal places).

If there are any specific discrepancies you would like to address or clarify further, please let me know!