Exercise 29

When people buy a large ticket item like a car or a house, they often take out a loan to make the purchase. The loan is paid back in monthly installments until the entire amount of the loan, plus interest, is paid. The monthly payment that the borrower has to make depends on the amount $P$ of money borrowed (called the principal), the duration $t$ of the loan in years, and the interest rate $r$. For example, if we borrow $18,000 to buy a car, the monthly payment $M$ that we need to make to pay off the loan is given by the formula $$\begin{gathered} M(r,t) = \frac{1500r}{1-\frac{1}{\left(1+\frac{r}{12}\right)^{12t}}}. \end{gathered}$$

1. Find the monthly payments on this loan if the interest rate is 6% and the duration of the loan is 5 years.

2. Create a table of values that illustrates the trace of $M$ with $r$ fixed at 5%. Use yearly values of $t$ from 2 to 6. Round payments to the nearest penny. Explain in detail in words what this trace tells us about $M$.

3. Create a table of values that illustrates the trace of $M$ with $t$ fixed at 3 years. Use rates from 3% to 11% in increments of 2%. Round payments to the nearest penny. Explain in detail what this trace tells us about $M$.

4. Consider the combinations of interest rates and durations of loans that result in a monthly payment of $200. Solve the equation $M(r,t) = 200$ for $t$ to write the duration of the loan in terms of the interest rate. Graph this level curve and explain as best you can the relationship between $t$ and $r$.