Question

The model \(t=12.542 \ln \left(\frac{x}{x-1000}\right), \quad x>1000\) approximates the length of a home mortgage of \(\$ 150,000\) at \(8 \%\) interest in terms of the monthly payment. In the model, \(t\) is the length of the mortgage (in years) and \(x\) is the monthly payment (in dollars) (see figure). (a) Use the model to approximate the length of a \(\$ 150,000\) mortgage at \(8 \%\) interest when the monthly payment is \(\$ 1100.65\) and when the monthly payment is \(\$ 1254.68\). (b) Approximate the total amount paid over the term of the mortgage with a monthly payment of \(\$ 1100.65\) and with a monthly payment of \(\$ 1254.68\). (c) Approximate the total interest charge for a monthly payment of \(\$ 1100.65\) and for a monthly payment of \(\$ 1254.68\) (d) What is the vertical asymptote of the model? Interpret its meaning in the context of the problem.

Short answer

Length of the mortgage with monthly payments of $1100.65 and $1254.68 are calculated using the logarithmic model. The total amount paid over the term of the mortgage as well as the total interest charged is then ascertained. The vertical asymptote of this model is at \( x = 1000 \), meaning that a borrower needs to pay at least $1000 per month to pay off the $150,000 loan at 8% interest.

Step-by-step solution

Calculate the Length of the Mortgage

We substitute the given monthly payments into the equation to solve for \( t \). For a monthly payment of $1100.65, \( t = 12.542 \ln \left(\frac{1100.65}{1100.65-1000}\right) \). Similarly, for a monthly payment of $1254.68, \( t = 12.542 \ln \left(\frac{1254.68}{1254.68-1000}\right) \). This will provide the length of the mortgages in years.

Calculate the Total Amount Paid

We multiply the number of years (from step 1) by 12 to get the total number of months, then multiply that by the monthly payment to get the total amount paid over the term of the mortgage.

Calculate Total Interest Charged

We subtract the original loan amount of $150,000 from the total amount paid (from step 2) to find the total interest charged over the life of the mortgage.

Identify the Vertical Asymptote

The vertical asymptote of the function is given by the line \(\ x = 1000\). This means that the minimum monthly payment the borrower can make to eventually pay off the loan is $1000.

Interpretation

An interpretation of the vertical asymptote is that if the monthly payment amount is less than the asymptote, the model predicts that it will require an infinite number of months to pay off the loan. This implies that any monthly payment less than $1000 would not be enough to eventually pay off the loan of $150,000 at an interest of 8%.

Key concepts

Understanding Loan Interest

Loan interest is the cost you pay for borrowing money from a lender. In the context of a mortgage, this interest is typically expressed as an annual percentage rate (APR). A mortgage operates as a type of amortizing loan, where each payment pays off a portion of the interest and a portion of the principal amount.

When you have an interest rate, like 8% in this problem, it means that the lender charges you 8% of the remaining principal balance as interest each year. Your mortgage payments are structured so that in the early years, a larger part of each payment goes towards interest, while in later years, more of the payment goes towards the principal.

Understanding the concept of loan interest is key because it influences how much you'll end up paying over the life of the loan. It is critical to consider when planning your finances for home buying, as higher interest rates mean higher total costs over the life of the loan.

Calculating Your Monthly Payment

The monthly payment is the amount you pay each month to the mortgage lender. It is calculated based on the loan amount, the interest rate, and the term of the loan. In mathematical terms, this involves solving the mortgage equation:

\[M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\]
where:

• \(M\) is the monthly payment,
• \(P\) is the principal loan amount (in this case \\(150,000),
• \(r\) is the monthly interest rate (annual rate divided by 12 months), and
• \(n\) is the total number of payments (loan term in years times 12).

Finding the correct monthly payment amount is crucial because it helps ensure that you are not overcommitting or underestimating your budget. When evaluating different monthly payment scenarios, like \)1100.65 or $1254.68, it affects the total cost and the time it takes to pay off the loan.

While the model uses a logarithmic function to estimate payments, ensuring the computations align with committing financially sustainable amounts can significantly impact your financial health.

Exploring the Vertical Asymptote

In the context of the given function, a vertical asymptote is represented at \(x = 1000\). In simple terms, an asymptote is a value that a function approaches but never actually reaches.

This means that in our mortgage model, if the monthly payment \(x\) is \(1000, the length of time required to pay off the mortgage becomes infinite. That's because at \)1000, you are essentially only covering the interest without reducing the principal loan amount.

Thus, a payment less than $1000 would not be sufficient to lower the principal over time, leading to a never-ending repayment scenario.

• Understanding the vertical asymptote helps borrowers realize the minimum payment needed to actually start reducing the principal.
• It acts as a cautionary check when determining how low your monthly payments can go while still effectively paying down your debt.

Recognizing the significance of this asymptote helps you avoid the trap of paying infinitely without progress, ensuring that your financial commitments are aligned with actual debt reduction.