$1600 principal earning 7%, compounded semi annually, after 33 years.
$4979.11
$14,920.54
$112,992.00
$15,494.70
The correct answer is $15,494.70.
To calculate the future value of the principal, we can use the formula for compound interest:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
A = future value of the principal
P = initial principal
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
Plugging in the given values:
P = $1600
r = 7% = 0.07
n = 2 (compounded semiannually)
t = 33
\[A = 1600 \left(1 + \frac{0.07}{2}\right)^{2 \cdot 33}\]
\[A = 1600 \left(1 + 0.035\right)^{66}\]
\[A = 1600 \cdot 1.035^{66}\]
\[A = 1600 \cdot 8.820271\]
\[A = 14,112.4336\]
Therefore, after 33 years, the principal will grow to $14,112.43. However, this answer is not one of the options provided.
It seems that there might be an error in the answer choices, as none of them match the calculated future value.
To calculate the future value of the principal, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Future value
P = Principal amount
r = Interest rate
n = Number of times compounded per year
t = Number of years
Given:
P = $1600
r = 7% (or 0.07)
n = 2 (semi-annually)
t = 33 years
Substitute the given values into the formula:
A = 1600(1 + 0.07/2)^(2*33)
A = 1600(1 + 0.035)^(66)
A = 1600(1.035)^66
A ≈ $14,920.54
Therefore, the answer is $14,920.54.
To calculate the future value of a principal amount compounded semiannually, we use the formula:
A = P(1 + r/n)^(nt)
Where:
A = future value
P = principal amount
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, the principal amount is $1600, the annual interest rate is 7% (or 0.07 as a decimal), the number of times interest is compounded per year is 2 (semiannually), and the number of years is 33.
Plugging in these values to the formula, we get:
A = 1600(1 + 0.07/2)^(2*33)
First, let's calculate the exponent part: (1 + 0.07/2)^(2*33) = (1 + 0.035)^66
Now, let's calculate the future value:
A = 1600 * (1.035)^66
A ≈ $112,992.00
Therefore, the correct answer is $112,992.00.