Answer :
Step-by-step explanation:
To determine the length of time the loan was taken out, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the total amount repaid (loan amount + interest)
P is the principal amount (loan amount)
r is the annual interest rate (in decimal form)
n is the number of times interest is compounded per year
t is the time period in years
Given:
P = $9000.00 (principal amount)
A = $9000.00 + $3728.00 = $12728.00 (total amount repaid)
r = 9.4% = 0.094 (annual interest rate in decimal form)
n = 4 (compounded quarterly)
We need to solve for t, the time period in years.
Substituting the given values into the formula, we have:
$12728.00 = $9000.00(1 + 0.094/4)^(4t)
Dividing both sides by $9000.00, we get:
1.414222222 = (1.0235)^(4t)
Taking the natural logarithm (ln) of both sides to eliminate the exponent:
ln(1.414222222) = ln(1.0235)^(4t)
Using logarithm properties, we can bring down the exponent:
ln(1.414222222) = 4t * ln(1.0235)
Now we can solve for t by dividing both sides by 4 * ln(1.0235):
t = ln(1.414222222) / (4 * ln(1.0235))
Using a calculator, we find:
t ≈ 6.25
Therefore, the loan was taken out for approximately 6.25 years.
