Suppose that $2000 is loaned at a rate of 6%, compounded monthly. Assuming that no payments are made, find the amount owed after 4 years.Do not round any intermediate computations, and round your answer to the nearest cent.

To find the amount owed after 4 years for a loan of $2000 at an interest rate of 6%, compounded monthly, we can use the compound interest formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount (the initial amount of money, which is $2000).
• $r$ is the annual interest rate (decimal) (6% or 0.06).
• $n$ is the number of times that interest is compounded per year (monthly, so 12 times a year).
• $t$ is the time the money is invested for in years (4 years).

Plugging in the values:

$P = 2000$ $r = 0.06$ $n = 12$ $t = 4$

The formula becomes:

$A = 2000 \left(1 + \frac{0.06}{12}\right)^{12 \times 4}$

First, calculate the monthly interest rate:

$\frac{0.06}{12} = 0.005$

Next, calculate the exponent:

$12 \times 4 = 48$

Now, substitute these values back into the formula:

$A = 2000 \left(1 + 0.005\right)^{48}$ $A = 2000 \left(1.005\right)^{48}$

Calculate $(1.005)^{48}$:

$(1.005)^{48} \approx 1.28368$

Finally, multiply by the principal amount:

$A = 2000 \times 1.28368$ $A \approx 2567.36$

So, the amount owed after 4 years, rounded to the nearest cent, is $2567.36.