91Ó°ÊÓ

How much can I borrow? The function in Example \(1.2\) can be rearranged to show the amount of money \(P=P(M, r, t)\), in dollars, that you can afford to borrow at a monthly interest rate of \(r\) (as a decimal) if you are able to make \(t\) monthly payments of \(M\) dollars: $$ P=M \times \frac{1}{r} \times\left(1-\frac{1}{(1+r)^{r}}\right) . $$ Suppose you can afford to pay \(\$ 350\) per month for 4 years. a. How much money can you afford to borrow for the purchase of a car if the prevailing monthly interest rate is \(0.75 \%\) ? (That is \(9 \%\) APR.) Express the answer in functional notation, and then calculate it. b. Suppose your car dealer can arrange a special monthly interest rate of \(0.25 \%\) (or \(3 \%\) APR). How much can you afford to borrow now? c. Even at \(3 \%\) APR you find yourself looking at a car you can't afford, and you consider extending the period during which you are willing to make payments to 5 years. How much can you afford to borrow under these conditions?

Step by step solution

01

Identify Variables and Constants for Part (a)

For part (a), we know that the monthly payment \(M = \$350\) and the payment period \(t\) is 4 years. Since there are 12 months per year, this means \(t = 4 \times 12 = 48\) months. The monthly interest rate \(r = 0.75\% = 0.0075\). We need to rearrange the given formula into functional notation and calculate \(P\).

02

Function Notation for Part (a)

Express the amount of money \(P\) you can afford to borrow in function notation based on the given formula. The function will be \(P(M, r, t) = M \times \frac{1}{r} \times \left(1 - \frac{1}{(1+r)^t}\right)\). Substitute the known values to calculate \(P\).

03

Calculate Amount for Part (a)

Substitute \(M = 350\), \(r = 0.0075\), and \(t = 48\) into the function:\[P(350, 0.0075, 48) = 350 \times \frac{1}{0.0075} \times \left(1 - \frac{1}{(1+0.0075)^{48}}\right)\]Calculate using a calculator:\[P \approx 350 \times 133.3333 \times (1 - 0.6954) = 350 \times 133.3333 \times 0.3046 \approx 14,204\].

04

Identify Variables and Constants for Part (b)

For part (b), the monthly interest rate \(r\) is reduced to \(0.25\% = 0.0025\), with \(M = 350\) and \(t = 48\) remaining the same. Use the same function \(P(M, r, t)\) and substitute the new interest rate to calculate \(P\).

05

Calculate Amount for Part (b)

Substitute \(M = 350\), \(r = 0.0025\), and \(t = 48\) into the function:\[P(350, 0.0025, 48) = 350 \times \frac{1}{0.0025} \times \left(1 - \frac{1}{(1+0.0025)^{48}}\right)\]Calculate using a calculator:\[P \approx 350 \times 400 \times (1 - 0.88849) = 350 \times 400 \times 0.11151 \approx 15,611\].

06

Identify Variables for Part (c)

For part (c), the monthly interest rate \(r = 0.0025\), the monthly payment \(M = 350\), and the payment period \(t\) is extended to 5 years, so \(t = 5 \times 12 = 60\). Use the same function to find \(P\).

07

Calculate Amount for Part (c)

Substitute \(M = 350\), \(r = 0.0025\), and \(t = 60\) into the function:\[P(350, 0.0025, 60) = 350 \times \frac{1}{0.0025} \times \left(1 - \frac{1}{(1+0.0025)^{60}}\right)\]Calculate using a calculator:\[P \approx 350 \times 400 \times (1 - 0.86435) = 350 \times 400 \times 0.13565 \approx 18,991\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Loan Calculations

Loan calculations are essential for understanding how much money you can borrow based on your ability to make monthly payments. Let's say you want to buy a car and can pay \(350 monthly for it. To calculate how much you can borrow, we use a specific formula:

• Amount you can borrow (\(P\))
• Monthly payment (\(M\))
• Monthly interest rate (\(r\))
• Number of payments (\(t\))

The loan formula is:\[P(M, r, t) = M \times \frac{1}{r} \times \left(1 - \frac{1}{(1+r)^t}\right)\]This formula helps determine \(P\), or the total amount you can afford to borrow. In the problem example, your monthly payment is fixed at \)350, and the period of payment, \(t\), is derived by converting years into months. Adjust the formula inputs depending on the interest rate and duration your lender offers.

Interest Rates

Interest rates play a crucial role in determining how much you can borrow. When calculating loans, even a small change in the interest rate can have a significant effect. Let's explore how this works:

• The monthly interest rate ($r$) is a decimal form of the annual percentage rate (APR) divided by 12. For example, a 9% APR means a 0.75% monthly rate.
• At a higher interest rate, you can borrow less because more of your monthly payment goes towards paying interest.
• When the interest rate is lower, such as 3% APR, represented as 0.25% monthly, you can borrow more because less of your payment goes toward interest.

In our example, you can borrow $14,204 at a 9% APR, but $15,611 at a reduced 3% APR, assuming all other loan terms remain the same.

Mathematical Modeling

Mathematical modeling involves creating equations to solve real-world problems like loan calculations. It involves using functions to predict outcomes based on different variables. Here's how it works with our loan example:

• Identify variables: Monthly payment, interest rate, and payment period.
• Use a function, like the one provided, to represent the relationship between these variables. This function helps us compute the result when variables change.
• Test different scenarios: Changing either the interest rate or the number of payments affects how much you can borrow.

This approach allows you to anticipate how different loan terms change what you can afford. Expanding the payment period to 5 years increases the amount you can borrow - it's now $18,991 at a 3% APR. Mathematical modeling gives you the tools to make informed financial decisions.