### Estimate the maximum interest rate that you can afford

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Question - Suppose you take a loan of P dollars and plan to pay it back in monthly payments over Y years with an annual interest rate of p%. Letting r = p/100 (for example, if p = 8, then r = 0.08), your monthly payments as a function of r will be m(r) = P + (r/12)/(1-(1+(r/12))-12Y)

1. Assume your loan amount is P = \$20,000 and you pay it back over Y = 10 years. Graph the monthly payment function m(r) for 0 ≤ r ≤ 0.2. How do the monthly payments change as r increases?

2. With p = \$20,000 and Y = 10, suppose you can afford monthly payments of at most \$225. Using the graph of Step 1, estimate the maximum interest rate that you can afford.

3. Assume your loan amount is P = \$20,000 and you pay it back over Y = 5 years. Graph the monthly payment function m(r) for 0 ≤ r ≤ 0.2. How do the monthly payments for Y = 5 and Y = 10 compare for the same values of r? Explain.

4. Make graphs of the monthly payment function with P = \$20,000 and Y = 5, 10, and 20 years. Describe how the monthly payments vary with Y for a fixed interest rate. Explain this behavior.

5. By graphing the monthly payment function for various values of P and Y, complete the following sentence: For the same interest rate and monthly payments, one could borrow \$20,000 and pay it back over 10 years or borrow \$ ____ and pay it back over 15 years.

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