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.