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Chapter 4: Problem 55

Some lending institutions calculate the monthly payment \(M\) on a loan of \(L\) dollars at an interest rate \(r\) (expressed as a decimal) by using the formula $$M=\frac{L r k}{12(k-1)}$$ where \(k=[1+(r / 12)]^{12 t}\) and \(t\) is the number of years that the loan is in effect. Car loan An automobile dealer offers customers no-downpayment 3-year loans at an interest rate of \(10 \% .\) If a customer can afford to pay 500 dollars per month, find the price of the most expensive car that can be purchased.

Short Answer

Expert verified
First, identify what each variable in the formula represents. Here, \(M\) stands for the monthly payment, \(L\) is the loan amount (price of the car), \(r\) is the annual interest rate as a decimal, \(t\) is the number of years the loan is for, and \(k\) is a factor defined as \(k = \left[1 + \left(\frac{r}{12}\right)\right]^{12t}\).

Step by step solution

01

Understand the Variables

First, identify what each variable in the formula represents. Here, \(M\) stands for the monthly payment, \(L\) is the loan amount (price of the car), \(r\) is the annual interest rate as a decimal, \(t\) is the number of years the loan is for, and \(k\) is a factor defined as \(k = \left[1 + \left(\frac{r}{12}\right)\right]^{12t}\).

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Key Concepts

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

Interest Rate
In financial terms, an interest rate is the proportion of a loan that is charged as interest to the borrower. It is usually expressed as a percentage. For example, if you take out a car loan with an annual interest rate of 10%, you are being charged 10% of the remaining balance annually until the loan is paid off. In the loan payment formula, knowing the interest rate is essential because it affects the total cost of borrowing money.

Remember, when working with formulas, it's crucial to convert the percentage interest rate into a decimal. To do this, divide the percentage by 100. For instance, 10% becomes 0.10. You'll also need to consider whether the interest is applied monthly, as indicated by \( r / 12 \) in calculations.
  • The annual interest rate is divided by 12 since it's compounded monthly.
  • As a decimal, it simplifies processing within mathematical formulas.
The smaller the interest rate, the less you'll pay over the life of the loan. Understanding this helps effectively budget loan payments.
Monthly Payment Formula
To calculate monthly payments on loans, lenders use a specific formula \( M = \frac{L r k}{12(k-1)} \) where:
  • \( M \) is the monthly payment.
  • \( L \) is the total loan amount or the cost of the car.
  • \( r \) is the annual interest rate expressed as a decimal (i.e., 10% becomes 0.10).
  • \( k \) is a key factor defined as \( k = \left[1 + \left(\frac{r}{12}\right)\right]^{12t} \).
This formula balances the need to repay the loan and the accruing interest. The role of \( k \) here is particularly important as it adjusts for the impact of compounding interest across the loan term.

Understanding this formula allows you to explore different loan scenarios:
  • Altering the interest rate or loan amount gives different monthly payments.
  • It enables precise financial planning by visualizing payment impact if conditions change.
Use this tool to determine how much you can afford monthly when considering new loans.
Loan Term
The loan term refers to the length of time you're given to repay a loan. It's impacted significantly by the time period indicated as \( t \) in the formula, representing years. Depending on the loan type, terms can range from short (a few months) to long term (up to 30 years or more).

The choice of loan term largely determines how much interest you will pay over the duration of the loan. Longer terms generally lead to higher total interest paid, while shorter terms can mean higher monthly payments but less total interest. Here are a few important considerations:
  • A 3-year term, as in this car loan example, means the loan will be paid off in 36 payments.
  • Loan terms determine \( k \) because \( k = \left[1 + \left(\frac{r}{12}\right)\right]^{12t} \), and \( t \) is used here to represent years.
  • Longer loan terms spread out payments, often lowering each monthly bill but increasing interest paid.
Deciding on a loan term that fits your financial situation will help manage both the short-term and long-term budget effectively.

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Most popular questions from this chapter

In \(1974,\) Johnny Miller won 8 tournaments on the PGA tour and accumulated 353,022 dollars in official season earnings. In \(1999,\) Tiger Woods accumulated 6,616,585 dollars with a similar record. (a) Suppose the monthly inflation rate from 1974 to 1999 was \(0.0025(3 \% / \mathrm{yr}) .\) Use the compound interest formula to estimate the equivalent value of Miller's winnings in the year \(1999 .\) Compare your answer with that from an inflation calculation on the web (e.g., bls.gov/cpi/home.htm). (b) Find the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings. (c) What type of function did you use in part (a)? part (b)?

Exer. \(51-54\) : Chemists use a number denoted by \(p H\) to describe quantitatively the acidity or basicity of solutions. By definition, \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right],\) where \(\left[\mathbf{H}^{+}\right]\) is the hydrogen ion concentration in moles per liter. Approximate the \(\mathrm{pH}\) of each substance. (a) vinegar: \(\left[\mathrm{H}^{+}\right] \approx 6.3 \times 10^{-3}\) (b) carrots: \(\left[\mathrm{H}^{+}\right] \approx 1.0 \times 10^{-5}\) (c) sea water: \(\left[\mathrm{H}^{+}\right] \approx 5.0 \times 10^{-9}\)

In 1840 , Britain experienced a bovine (cattle and oxen) epidemic called epizooty. The estimated number of new cases every 28 days is listed in the table. At the time, the London Daily made a dire prediction that the number of new cases would continue to increase indefinitely. William Farr correctly predicted when the number of new cases would peak. Of the two functions $$\begin{array}{l}f(t)=653(1.028)^{t} \\\g(t)=54,700 e^{-(t-200)^{2} / 7500}\end{array}$$ and one models the newspaper's prediction and the other models Farr's prediction, where \(t\) is in days with \(t=0\) corresponding to August \(12,1840\). $$\begin{array}{|c|c|}\hline \text { Date } & \text { New cases } \\\\\hline \text { Aug. 12 } & 506 \\\\\hline \text { Sept. 9 } & 1289 \\\\\hline \text { Oct. 7 } & 3487 \\\\\hline \text { Nov. 4 } & 9597 \\\\\hline \text { Dec. 2 } & 18,817 \\\\\hline \text { Dec. 30 } & 33,835 \\\\\hline \text { Jan. 27 } & 47,191 \\\\\hline\end{array}$$ (a) Graph each function, together with the data, in the viewing rectangle \([0,400,100]\) by \([0,60,000,10,000]\) (b) Determine which function better models Farr's prediction. (c) Determine the date on which the number of new cases peaked.

Ventilation is an effective way to improve indoor air quality. In nonsmoking restaurants, air circulation requirements (in \(\mathrm{ft}^{3} / \mathrm{min}\) ) are given by the function \(V(x)=35 x,\) where \(x\) is the number of people in the dining area. (a) Determine the ventilation requirements for 23 people. (b) Find \(V^{-1}(x) .\) Explain the significance of \(V^{-1}\) (c) Use \(V^{-1}\) to determine the maximum number of people that should be in a restaurant having a ventilation capability of \(2350 \mathrm{ft}^{3} / \mathrm{min}\)

Coal consumption A country presently has coal reserves of 50 million tons. Last year 6.5 million tons of coal was consumed. Past years' data and population projections suggest that the rate of consumption \(R\) (in million tons/year) will increase according to the formula \(R=6.5 e^{0.02 t}\) and the total amount \(T\) (in million tons) of coal that will be used in t years is given by the formula \(T=325\left(e^{0.02 t}-1\right) .\) If the country uses only its own resources, when will the coal reserves be depleted?
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