Problem 56 - Precalculus

Question

Amy has a credit card debt in the amount of \(\$ 12,000 .\) The annual interest is \(18 \% .\) Her time \(t\) to pay off the loan is given by $$t=-\frac{\ln \left[1-\frac{12,000(0.18)}{n R}\right]}{n \ln \left(1+\frac{0.18}{n}\right)}$$ where \(n\) is the number of payment periods per year and \(R\) is the periodic payment. a. Use a graphing utility to graph $$t_{1}=-\frac{\ln \left[1-\frac{12,000(0.18)}{12 x}\right]}{12 \ln \left(1+\frac{0.18}{12}\right)} \text { as } Y_{1} \text { and }$$ $$t_{2}=-\frac{\ln \left[1-\frac{12,000(0.18)}{26 x}\right]}{26 \ln \left(1+\frac{0.18}{26}\right)} \text { as } Y_{2}$$ Explain the difference in the two graphs. b. Use the \([\text { TRACE }]\) key to estimate the number of years that it will take Amy to pay off her credit card if she can afford a monthly payment of \(\$ 300 .\) c. If she can make a biweekly payment of \(\$ 150,\) estimate the number of years that it will take her to pay off the credit card. d. If Amy adds \(\$ 100\) more to her monthly or \(\$ 50\) more to her biweekly payment, estimate the number of years that it will take her to pay off the credit card.

Step-by-Step Solution

Answer

More frequent payments (biweekly) generally reduce the payoff time due to more frequent interest reduction, while increasing payment amounts further hastens debt repayment.

Step 1: Setting Up the Functions

First, let's understand the given functions for credit card debt repayment in terms of number of periods per year and payment amounts.1. **Monthly Payments:** For the monthly payments, Amy pays 12 times per year (because monthly payments are annual payments divided into 12 periods). So, the formula becomes: \[ t_1 = -\frac{\ln \left[1-\frac{12,000(0.18)}{12x}\right]}{12 \ln \left(1+\frac{0.18}{12}\right)} \]2. **Biweekly Payments:** For biweekly payments, Amy pays 26 times per year (there are about 26 two-week periods in a year). So, the formula becomes: \[ t_2 = -\frac{\ln \left[1-\frac{12,000(0.18)}{26x}\right]}{26 \ln \left(1+\frac{0.18}{26}\right)} \]

Step 2: Graph the Functions Using a Graphing Utility

Use a graphing calculator or software to plot the given equations. Set the x-axis to represent the periodic payment amount (\(x\)), and the y-axis to represent the time in years (\(t\)). - **Graph \(Y_1:\)** Represents the monthly payment calculation.- **Graph \(Y_2:\)** Represents the biweekly payment calculation.Both graphs should display the relationship between payment amounts and time taken to pay off the debt.

Step 3: Analyzing the Graphs for Payment Difference

Upon plotting the two functions, observe:- **\(Y_1 (Monthly Payments):\)** Represents payment periods divided into larger chunk payments that are less frequent but larger.- **\(Y_2 (Biweekly Payments):\)** Represents more frequent payments of smaller amounts.The difference in the graphs lies in how the payment frequency affects the time to payoff. The biweekly payment graph generally reduces the repayment period because of more frequent interest repayments.

Step 4: Estimate for Monthly Payment of $300

Using the graph for \(Y_1\), trace along the curve for a monthly payment of \(\\(300\). Observe where the line intersects with the graph and note down the value of \(t\). This is where \(x = 300\).The trace will provide an approximate number of years for Amy to pay off her debt under a \(\\)300\) monthly payment schedule.

Step 5: Estimate for Biweekly Payment of $150

Using the graph for \(Y_2\), trace along the curve for a biweekly payment of \(\\(150\). Find the intersection point on the graph to determine the value of \(t\). This is where \(x = 150\) given 26 payments per year.The trace here gives the approximate time it will take for Amy to pay off her credit card with a \(\\)150\) biweekly payment.

Step 6: Estimate for Increased Monthly Payment

For an increased monthly payment of \(\\(400\), track \(Y_1\) at \(x = 400\) using the TRACE function on the graph. This will show how much faster the loan will be paid off with an additional \(\\)100\) to the monthly payments.

Step 7: Estimate for Increased Biweekly Payment

For an increased biweekly payment of \(\\(200\), use the TRACE function on \(Y_2\) at \(x = 200\). This gives the new repayment time with the additional \(\\)50\) per biweekly payment.

Key Concepts

Logarithmic Functions

Logarithmic functions are crucial when it comes to solving equations that involve exponents, especially in financial calculations. The logarithmic function is the inverse of the exponential function. It helps us reverse the process of exponentiation, making it easier to determine the time required to achieve a particular value in processes characterized by constant growth or decay, such as compound interest.

For example, in the formula used by Amy to determine her debt repayment time, the base of the logarithm is related to the growth of debt due to interest rates. It essentially helps in finding how long it will take to reach zero debt given a certain payment schedule by reversing the exponential growth of the debt due to interest.

• Logarithms come in two common bases: base 10 (common logarithm) and base \(e\) (natural logarithm).
• Natural logarithms (\

Graphing Calculators

Graphing calculators are powerful tools used to visualize mathematical functions and solve complex equations efficiently. They play a vital role when you need to analyze functions like those in Amy’s debt repayment problem. These calculators allow you to input equations and observe the graphical representation of functions to better understand how variables interact.

For Amy's problem, a graphing calculator helps by plotting the relationship between her payment amounts and the number of years required to pay off her debt. When you create graphs such as \(Y_1\) and \(Y_2\), the x-axis represents the payment amount, while the y-axis shows the time in years. By using a graphing calculator, you can easily determine how changes in payment amounts affect the time needed to be debt-free.

• Graphing calculators offer functions like \

Interest Rates

Interest rates significantly impact financial calculations, particularly in loans and credit card debts. They represent the cost of borrowing or the gain on savings, and are typically expressed as annual percentages. In Amy's scenario, the 18% annual interest can severely affect how long it takes to eliminate credit card debt.

Compounded interest, which is common in loans or credit balances, means that Amy needs to understand how interest accumulates over each period before her payment is made. If the interest is compounded monthly, it affects her total debt balance monthly, increasing how much she owes over time if she doesn't pay adequate amounts. Therefore, it makes her credit card debt increase exponentially if payments are less frequent or too low.

• Compound interest leads to exponential debt growth, making it crucial to prioritize paying off debts with high-interest rates.
• Understanding the effect of interest rates is critical in making informed financial decisions and planning effective strategies to alleviate debts or maximize investment returns.

Credit Card Debt Calculation

Calculating credit card debt payoff involves understanding how payments reduce principal over time, considering interest rates. With Amy's debt, the credit card payoff time is computed using specific formulas that integrate payment frequency and amount, which are vital to determine how quickly she can pay down her debt.

The formula used encompasses logarithmic functions to help calculate the payoff period given an annual interest rate, number of payment periods, and payment amount. Amy's scenario requires differentiating between monthly versus biweekly payments to see the effect of increased payment frequency on reducing debt faster.

• For a monthly payment, such as \\(300, the calculation estimates how many years are required at her current interest rate and debt level.
• Switching to biweekly payments at \\)150 per installment can significantly change the payoff timeline due to more frequent payments reducing the principal balance more often.
• Adding extra payments, like an additional \$100 monthly, can further expedite debt repayment by decreasing the outstanding balance quicker, thereby lessening interest accrued.

Understanding these calculations is essential for creating a manageable plan to stay ahead of credit card debt and avoid prolonged financial burden.