Question

The term \(t\) (in years) of a \(\$ 200,000\) home mortgage at \(7.5 \%\) interest can be approximated by \(t=-13.375 \ln \frac{x-1250}{x}, x>1250\) where \(x\) is the monthly payment in dollars. (a) Use a graphing utility to graph the model. (b) Use the model to approximate the term of a home mortgage for which the monthly payment is \(\$ 1398.43 .\) What is the total amount paid? (c) Use the model to approximate the term of a home mortgage for which the monthly payment is \(\$ 1611.19 .\) What is the total amount paid? (d) Find the instantaneous rate of change of \(t\) with respect to \(x\) when \(x=\$ 1398.43\) and \(x=\$ 1611.19\). (e) Write a short paragraph describing the benefit of the higher monthly payment.

Short answer

The total amounts paid for monthly payments of $1398.43 and $1611.19 will have to be calculated after we have their respective terms. The instantaneous rates of change provide insight into how quickly the term of the mortgage is decreasing as the monthly payment increases. The benefit of the higher monthly payment can involve shorter mortgage term and less total interest paid over the life of the loan.

Step-by-step solution

Execution of Part (b)

In this part, we need to find the term of the mortgage when the monthly payment is $1398.43. We plug this value into the given equation: \( t = -13.375 \ln \frac{1398.43-1250}{1398.43} \). After computing this, we get the term \( t \). The total amount paid can then be calculated by multiplying the monthly payment by the number of months in the term (remember to convert years to months).

Execution of Part (c)

Similar to part (b), we calculate the term of the mortgage with the monthly payment of $1611.19 using the given equation. The total amount paid will be calculated in the same way, by multiplying the monthly payment by the number of months (converted from years) in the term.

Execution of Part (d)

To find the instantaneous rate of change of \( t \) with respect to \( x \) , we need to differentiate the given equation with respect to \( x \), this will give us the derivative of \( t \) with respect to \( x \). Then, we substitute \( x = 1398.43 \) and \( x = 1611.19 \) into the derivative to find the instantaneous rates of change.

Execution of Part (e)

The comparison of the total amounts paid under the two different monthly payments will be done and also the benefits of higher monthly payment like reducing the term of the mortgage and total amount of interest paid over the life of the loan will be highlighted.

Key concepts

interest rate

A mortgage calculation deeply relies on understanding interest rates. The interest rate is the percentage of the loan amount that you, as a borrower, will pay to the lender in addition to repaying the principal amount. In our formula, the 7.5% interest reflects the cost of borrowing the money.
This interest rate affects the overall cost of the mortgage. With a higher interest rate, the monthly payments are increased, which in turn can reduce or extend the mortgage term.
Lower rates typically mean lower monthly payments and less total interest paid over the life of the loan. This is crucial in helping you determine the affordability and total cost of a mortgage. Having a grasp on how this percentage interacts with your principal amount can guide you in selecting the best mortgage offer.
You should ensure you understand how interest rates are calculated, as they can significantly affect your budget and long-term financial health.

logarithmic functions

Logarithmic functions are essential in this mortgage exercise as they appear in the formula used to calculate the mortgage term. A logarithm, in simplest terms, helps to break down multiplication into smaller, more manageable components using exponents.
In our case, the equation involves a natural logarithm (ln), specifically to express the term of the mortgage in relation to the monthly payments. This is because logarithms are particularly useful in modeling scenarios involving exponential decay or growth, such as the repayment structure of a mortgage.
The logarithmic model indicates how sensitive the term of the mortgage is to changes in the monthly payment. A slight increase in monthly payments logs a significant decrease in the mortgage term, thereby shortening the loan by a notable amount. Understanding logarithms helps you to appreciate how small differences in payment strategies can greatly impact the overall mortgage duration.

instantaneous rate of change

The instantaneous rate of change of a function gives us insight into how a certain variable affects another variable at a specific point of interest. For a mortgage, understanding this rate can provide valuable insights into how changes in the monthly payment amount can influence the mortgage duration.
To find the instantaneous rate of change, we differentiate our mortgage formula with respect to the monthly payment. This involves finding the derivative. The resulting derivative tells us how the mortgage term changes as the monthly payment changes. Inputting different values of monthly payments into this derivative provides specific rates at which the mortgage term changes.
This concept is highly beneficial as it allows the borrower to visualize the impact of adjusting payments. Recognizing how small changes in monthly payments can either increase or decrease the payment period is crucial for optimal financial planning and reducing total interest costs over the loan’s life.

monthly payments

Monthly payments play a central role in determining the term and total cost of a mortgage. The given formula involves the monthly payment amount, which significantly affects both the duration and the expenses incurred throughout the mortgage's life.
Choosing a higher monthly payment means you'll pay off the mortgage quicker, hence fewer interest payments over time. This can lead to substantial savings, as demonstrated by comparing two different payment scenarios, such as monthly payments of $1398.43 and $1611.19.
The difference between various payment amounts shows the relationship between payment size, debt term, and overall cost. A low monthly payment might seem attractive initially, but it could lead to a longer term and more interest paid. Balancing your monthly payment with your financial capabilities ensures you are getting the best deal on your mortgage. Consider how these trade-offs align with your financial goals and stability for the most economical approach to home ownership.