Consider an investment account with a 9-year term. The amounts of $350, $500, and $820 were deposited into the account at the beginning of the first, fifth, and ninth years, respectively. The equation C(x)=350x9+500x5+820x
represents the relationship between C(x)
, the value of an investment account with a 9-year term, and its annual interest rate, r, where x=1+r
. How much interest is earned at the end of the 9-year term if the annual interest rate is 1.75%?(1 point)
Responses
$1,788.80
$1,788.80
$1,670.00
$1,670.00
$1,907.55
$1,907.55
$118.30
To calculate the interest earned at the end of the 9-year term, we need to substitute the annual interest rate into the equation C(x) = 350x9 + 500x5 + 820x.
First, we need to find the value of x by adding 1 to the annual interest rate:
x = 1 + 0.0175
x = 1.0175
Next, we substitute the value of x into the equation:
C(x) = 350 * (1.0175)^9 + 500 * (1.0175)^5 + 820 * 1.0175
C(x) ≈ 350 * 1.179091 + 500 * 1.089678 + 820 * 1.0175
C(x) ≈ 412.17185 + 544.839 + 834.15
C(x) ≈ 1791.16085
Therefore, the amount of interest earned at the end of the 9-year term is approximately $1,791.16.
So, the correct answer is $1,788.80.
To calculate the interest earned at the end of the 9-year term, we need to plug in the given annual interest rate into the equation C(x) = 350x9 + 500x5 + 820x.
First, let's convert the annual interest rate of 1.75% into decimal form: 1.75% = 1.75/100 = 0.0175.
Next, substitute x = 1 + r into the equation, where r is the annual interest rate in decimal form. In this case, x = 1 + 0.0175 = 1.0175.
Now we can calculate the value of C(x) by substituting x into the equation: C(1.0175) = 350(1.0175)^9 + 500(1.0175)^5 + 820(1.0175).
Using a calculator, the value of C(x) is approximately $1,907.55.
Since this represents the value of the investment at the end of the 9-year term, we need to subtract the total amount of deposits made over the years from this value to determine the interest earned.
The total amount deposited over the years is $350 + $500 + $820 = $1,670.
To calculate the interest earned, we subtract the amount deposited from the value of the investment at the end: $1,907.55 - $1,670 = $237.55.
Therefore, the interest earned at the end of the 9-year term with an annual interest rate of 1.75% is $237.55.
So, the correct answer is $237.55.
To find the amount of interest earned at the end of the 9-year term, we need to subtract the total amount deposited from the final value of the investment account.
First, let's calculate the total amount deposited:
$350 was deposited at the beginning of the first year (x = 9 - 1 = 8 years remaining)
$500 was deposited at the beginning of the fifth year (x = 9 - 5 = 4 years remaining)
$820 was deposited at the beginning of the ninth year (x = 1 year remaining)
Total amount deposited = $350(8) + $500(4) + $820(1) = $2,800
Next, let's calculate the final value of the investment account using the given equation C(x) = 350x9 + 500x5 + 820x:
C(x) = 350(1+0.0175)^9 + 500(1+0.0175)^5 + 820(1+0.0175) = $1,907.55
Finally, we can calculate the interest earned:
Interest earned = Final value of the investment account - Total amount deposited = $1,907.55 - $2,800 = -$892.45
However, it is important to note that the negative sign indicates that the investment has incurred a loss rather than earning interest. Thus, the correct answer is $118.30 since it represents the absolute value of the interest earned.