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 $
.
To find the amount of interest earned, we need to subtract the sum of Kaira's deposits from the total value of the investment after 3 years.
The equation C(x) represents the total value of the investment after 3 years:
C(x) = 600x^3 + 600x^2 + 600x
To find the sum of Kaira's deposits, we need to find the sum of the first 3 terms of C(x) when x = 1 (since each deposit is made at the beginning of the year):
Sum of deposits = 600(1)^3 + 600(1)^2 + 600(1) = 600 + 600 + 600 = 1800
Now, to find the amount of interest earned, we subtract the sum of Kaira's deposits from the total value of the investment after 3 years:
Interest earned = C(x) - Sum of deposits = 600x^3 + 600x^2 + 600x - 1800
Given that x = 1 + r (where r is the interest rate), we can substitute this value into the equation:
Interest earned = 600(1 + r)^3 + 600(1 + r)^2 + 600(1 + r) - 1800
Simplifying this equation gives us:
Interest earned = 600(1 + 3r + 3r^2 + r^3) + 600(1 + 2r + r^2) + 600(1 + r) - 1800
Interest earned = 600 + 1800r + 1800r^2 + 600r^3 + 600 + 1200r + 600r^2 + 600 + 600r - 1800
Interest earned = 3600r^3 + 4200r^2 + 3600r
Now we can round the answer to two decimal places:
Interest earned = 3600(r^3 + r^2 + r)
Interest earned = 3600(0.0475^3 + 0.0475^2 + 0.0475)
Interest earned = 3600(0.002175 + 0.00225625 + 0.0475)
Interest earned = 3600(0.05193125)
Interest earned ≈ 185.75
Therefore, the amount of interest that Kaira will earn at the end of the 3-year term is approximately $185.75.
To find the amount of interest that Kaira will earn at the end of the 3-year term, we need to subtract the total value of her deposits from the total value of the investment after 3 years.
The equation C(x) represents the relationship between the value of the investment after 3 years and x, where x is 1 + the interest rate (r).
So, we can calculate the total value of the investment after 3 years using the equation C(x) = 600x^3 + 600x^2 + 600x, where x = 1 + r.
To find the difference between the total value of the investment and the sum of her $600 deposits, we need to subtract the sum of her deposits from the total value of the investment.
The sum of her deposits over 3 years is 600 + 600 + 600 = 1800.
Now, let's calculate the total value of the investment after 3 years:
C(x) = 600x^3 + 600x^2 + 600x
Substituting x = 1 + r:
C(1 + r) = 600(1 + r)^3 + 600(1 + r)^2 + 600(1 + r)
Simplifying:
C(1 + r) = 600(1 + 3r + 3r^2 + r^3) + 600(1 + 2r + r^2) + 600(1 + r)
C(1 + r) = 600 + 1800r + 1800r^2 + 600r^3 + 600 + 1200r + 600r^2 + 600 + 600r
C(1 + r) = 3600r^3 + 3000r^2 + 3600r + 2400
Now, let's calculate the difference between the total value of the investment and the sum of her deposits:
Difference = C(1 + r) - 1800
Difference = 3600r^3 + 3000r^2 + 3600r + 2400 - 1800
Difference = 3600r^3 + 3000r^2 + 3600r + 600
The amount of interest that Kaira will earn at the end of the 3-year term is the difference calculated above.
To compute this value, you would need to know the interest rate (r) that Kaira is earning on her investment.
To find the amount of interest that Kaira will earn at the end of the 3-year term, we need to calculate the difference between the total value of the investment after 3 years and the sum of her $600 deposits.
The equation that represents the relationship between the value of the investment after 3 years, C(x), is C(x) = 600x^3 + 600x^2 + 600x, where x = 1 + r and r is the interest rate.
To calculate the total value of the investment after 3 years, we can substitute x = 1 + 0.0475 (4.75% expressed as a decimal) into the equation:
C(1.0475) = 600(1.0475)^3 + 600(1.0475)^2 + 600(1.0475)
Simplifying the equation gives us:
C(1.0475) = 600(1.14290484) + 600(1.09765625) + 600(1.0475)
C(1.0475) = 685.74 + 658.59 + 628.5
C(1.0475) = 1972.83
The total value of the investment after 3 years is $1972.83.
Now, we need to calculate the sum of her $600 deposits:
Sum of deposits = $600 * 3
Sum of deposits = $1800
Finally, to find the amount of interest earned, we subtract the sum of deposits from the total value of the investment:
Interest earned = Total value of investment - Sum of deposits
Interest earned = $1972.83 - $1800
Interest earned = $172.83
Therefore, Kaira will earn $172.83 in interest at the end of the 3-year term.