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Chapter 2: Problem 10

If you borrow \(\$ 120,000\) at an APR of \(6 \%\) in order to buy a home, and if the lending institution compounds interest continuously, then your monthly payment \(M=M(Y)\), in dollars, depends on the number of years \(Y\) you take to pay off the loan. The relationship is given by $$ M=\frac{120000\left(e^{0.005}-1\right)}{1-e^{-0.06 Y}}. $$ a. Make a graph of \(M\) versus \(Y\). In choosing a graphing window, you should note that a home mortgage rarely extends beyond 30 years. b. Express in functional notation your monthly payment if you pay off the loan in 20 years, and then use the graph to find that value. c. Use the graph to find your monthly payment if you pay off the loan in 30 years. d. From part b to part \(\mathrm{c}\) of this problem, you increased the debt period by \(50 \%\). Did this decrease your monthly payment by \(50 \%\) ? e. Is the graph concave up or concave down? Explain your answer in practical terms. f. Calculate the average decrease per year in your monthly payment from a loan period of 25 to a loan period of 30 years.

Short Answer

Expert verified
Use the graph to determine payments: M(20) and M(30). Analyze changes and concavity from the graph.

Step by step solution

01

Set Up the Graph

To graph the relationship between monthly payment \(M\) and number of years \(Y\), set \(Y\) as the horizontal axis ranging from 0 to 30, as most mortgages do not exceed 30 years. Use the given equation \(M = \frac{120000\left(e^{0.005}-1\right)}{1-e^{-0.06 Y}}\) to plot \(M\) on the vertical axis.
02

Calculate M for 20 Years

Using the equation, find \(M\) when \(Y = 20\):\[M(20) = \frac{120000\left(e^{0.005}-1\right)}{1-e^{-0.06 \times 20}}.\]Calculate this value using a calculator to find the monthly payment for 20 years.
03

Determine M from Graph for 20 Years

Locate \(Y = 20\) on the graph you created in Step 1. The corresponding value of \(M\) on the vertical axis will give you the monthly payment for a 20-year mortgage.
04

Determine M from Graph for 30 Years

Locate \(Y = 30\) on the same graph, and find the value of \(M\) from the vertical axis. This gives the monthly payment for a 30-year mortgage.
05

Analyze 50% Increase in Loan Period

Compare the monthly payment values for 20 years and 30 years as determined in Steps 3 and 4. Check if the monthly payment for 30 years is 50% less than that for 20 years.
06

Determine Graph Concavity

Examine the shape of the graph. If the graph is bending upwards as \(Y\) increases, it is concave up; if bending downwards, it is concave down. Typically, decreasing rate of change in monthly payments indicates concave down.
07

Calculate Average Decrease per Year

Find the payment differences between 25 and 30 years from the graph. Divide the difference by the number of years to determine the yearly average decrease in monthly payments.

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Key Concepts

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

Continuously Compounded Interest
When you borrow money from a lending institution, the concept of interest is crucial in determining how much you will repay. Continuously compounded interest is a specific interest calculation method where interest is added instantaneously and constantly. This method is more frequent than even daily or monthly compounding and calculates interest as if it's growing at every microsecond.

For a given principal amount, the formula for continuously compounded interest is expressed as \( A = P \, e^{rt} \), where \( A \) is the amount of money accumulated after \( t \) years, \( P \) is the principal amount, \( r \) is the annual interest rate, and \( e \) is the base of the natural logarithm, approximately equal to 2.71828. This formula is foundational in finance, especially when considering the true cost of a loan over time, as seen in mortgages or investments.
  • Utilized for precision, capturing micro-moments of growth.
  • Higher than traditional compounding due to constant growth addition.
  • Important in finance for understanding true cost over time.
Graphing Functions
Graphing functions is a key mathematical skill that helps visualize relationships between variables. In our exercise, the function describes how monthly payments \( M \) change with the number of years \( Y \) you take to pay off a loan. Graphing this function provides insight into your financial obligations over time.

To create a meaningful graph, follow these steps:
  • Identify the independent variable (\( Y \) 鈥� years).
  • Mark it on the horizontal axis (x-axis).
  • Set the dependent variable (\( M \) 鈥� monthly payment).
  • Plot it on the vertical axis (y-axis).
  • Choose a suitable range for each axis; here, \( Y \) ranges from 0 to 30, reflecting typical mortgage lengths.
Utilize specific points like \( Y = 20 \) and \( Y = 30 \) to identify exact values on the graph. This visual representation simplifies the understanding of how different loan lengths impact financial obligations, showcasing trends like the change in monthly installments over time.
Loan Payments
Loan payments depend on several factors including principal amount, interest rate, and the term or length of the loan. In our exercise, the function \( M=M(Y) \) computes monthly payments based on these parameters over a continuous interest compounding model.

This is crucial because:
  • Lower monthly payments may result from longer loan terms like 30 years.
  • Longer terms lead to more interest paid over the loan's life despite lower monthly fees.
  • Calculating with continuously compounded interest offers a more accurate depiction of costs.
Understanding the balance between monthly payments and total interest paid is key in personal finance. Knowing how factors interact within the payment function helps make informed decisions regarding loan conditions that best fit personal or financial goals.
Function Notation
Function notation is an essential part of algebra, simplifying the representation and evaluation of functions. In your exercise, the payment function is denoted by \( M = M(Y) \), indicating that the monthly payment \( M \) is a function of the years \( Y \).

Function notation benefits include:
  • Clear communication of which variable is dependent (here, \( M \) depends on \( Y \)).
  • Simplifies substitution of values, like finding \( M(20) \) for a 20-year term.
  • Highlights the function relationship and simplifies complex equations for evaluation.
In mathematics, using function notation is a powerful tool for expressing computations clearly and exploring the relationships between different quantities. For students tackling algebra problems, mastering function notation enhances their ability to solve, interpret, and understand mathematical models.

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Most popular questions from this chapter

One interesting problem in the study of dinosaurs is to determine from their tracks how fast they ran. The scientist R. McNeill Alexander developed a formula giving the velocity of any running animal in terms of its stride length and the height of its hip above the ground. 7 The stride length of a dinosaur can be measured from successive prints of the same foot, and the hip height (roughly the leg length) can be estimated on the basis of the size of a footprint, so Alexander's formula gives a way of estimating from dinosaur tracks how fast the dinosaur was running. See Figure \(2.57 .\) If the velocity \(v\) is measured in meters per second, and the stride length \(s\) and hip height \(h\) are measured in meters, then Alexander's formula is $$ v=0.78 s^{1.67} h^{-1.17} . $$ (For comparison, a length of 1 meter is \(39.37\) inches, and a velocity of 1 meter per second is about \(2.2\) miles per hour.) a. First we study animals with varying stride lengths but all with a hip height of 2 meters (so \(h=2\) ). i. Find a formula for the velocity \(v\) as a function of the stride length \(s\). ii. Make a graph of \(v\) versus \(s\). Include stride lengths from 2 to 10 meters. iii. What happens to the velocity as the stride length increases? Explain your answer in practical terms. iv. Some dinosaur tracks show a stride length of 3 meters, and a scientist estimates that the hip height of the dinosaur was 2 meters. How fast was the dinosaur running? b. Now we study animals with varying hip heights but all with a stride length of 3 meters (so \(s=3\) ). i. Find a formula for the velocity \(v\) as a function of the hip height \(h\). ii. Make a graph of \(v\) versus \(h\). Include hip heights from \(0.5\) to 3 meters. iii. What happens to the velocity as the hip height increases? Explain your answer in practical terms.

A mole of a chemical compound is a fixed number, \({ }^{16}\) like a dozen, of molecules (or atoms in the case of an element) of that compound. A mole of water, for example, is about 18 grams, or just over a half an ounce in your kitchen. Chemists often use the mole as the measure of the amount of a chemical compound. A mole of carbon dioxide has a fixed mass, but the volume \(V\) that it occupies depends on pressure \(p\) and temperature \(T\); greater pressure tends to compress the gas into a smaller volume, whereas increasing temperature tends to make the gas expand into a larger volume. If we measure the pressure in atmospheres ( 1 atm is the pressure exerted by the atmosphere at sea level), the temperature in kelvins, and the volume in liters, then the relationship is given by the ideal gas law: $$ p V=0.082 T \text {. } $$ a. Solve the ideal gas law for the volume \(V\). b. What is the volume of 1 mole of carbon dioxide under \(3 \mathrm{~atm}\) of pressure at a temperature of 300 kelvins? c. Solve the ideal gas law for pressure. d. What is the pressure on 1 mole of carbon dioxide if it occupies a volume of \(0.4\) liter at a temperature of 350 kelvins? e. Solve the ideal gas law for temperature. f. At what temperature will 1 mole of carbon dioxide occupy a volume of 2 liters under a pressure of \(0.3 \mathrm{~atm}\) ?

For retailers who buy from a distributor or manufacturer and sell to the public, a major concern is the cost of maintaining unsold inventory. You must have appropriate stock to do business, but if you order too much at a time, your profits may be eaten up by storage costs. One of the simplest tools for analysis of inventory costs is the basic or der quantity model. It gives the yearly inventory expense \(E=E(c, N, Q, f)\) when the following inventory and restocking cost factors are taken into account: \- The carrying cost \(c\), which is the cost in dollars per year of keeping a single unsold item in your warehouse. \- The number \(N\) of this item that you expect to sell in 1 year. \- The number \(Q\) of items you order at a time. \- The fixed costs \(f\) in dollars of processing a restocking order to the manufacturer. (Note: This is not the cost of the order; the price of an item does not play a role here. Rather, \(f\) is the cost you would incur with any order of any size. It might include the cost of processing the paperwork, fixed costs you pay the manufacturer for each order, shipping charges that do not depend on the size of the order, the cost of counting your inventory, or the cost of cleaning and rearranging your warehouse in preparation for delivery.) The relationship is given by $$ E=\left(\frac{Q}{2}\right) c+\left(\frac{N}{Q}\right) f \text { dollars per year. } $$ A new-car dealer expects to sell 36 of a particular model car in the next year. It costs \(\$ 850\) per year to keep an unsold car on the lot. Fixed costs associated with preparing, processing, and receiving a single order from Detroit total \(\$ 230\) per order. a. Using the information provided, express the yearly inventory expense \(E=E(Q)\) as a function of \(Q\), the number of automobiles included in a single order. b. What is the yearly inventory expense if 3 cars at a time are ordered? c. How many cars at a time should be ordered to make yearly inventory expenses a minimum? d. Using the value of \(Q\) you found in part c, determine how many orders to Detroit will be placed this year. e. What is the average rate of increase in yearly inventory expense from the number you found in part \(\mathrm{c}\) to an order of 2 cars more?

The background for this exercise can be found in Exercises 11, 12, 13, and \(14 \mathrm{in} \mathrm{Sec}\) tion 1.4. A manufacturer of widgets has fixed costs of \(\$ 700\) per month, and the variable cost is \(\$ 65\) per thousand widgets (so it costs \(\$ 65\) to produce 1 thousand widgets). Let \(N\) be the number, in thousands, of widgets produced in a month. a. Find a formula for the manufacturer's total cost \(C\) as a function of \(N\). b. The highest price \(p\), in dollars per thousand widgets, at which \(N\) can be sold is given by the formula \(p=75-0.02 \mathrm{~N}\). Using this, find a formula for the total revenue \(R\) as a function of \(N\). c. Use your answers to parts a and \(b\) to find \(a\) formula for the profit \(P\) of this manufacturer as a function of \(N\). d. Use your formula from part c to determine the two break-even points for this manufacturer. Assume that the manufacturer can produce at most 500 thousand widgets in a month.

A breeding group of foxes is introduced into a protected area, and the population growth follows a logistic pattern. After \(t\) years the population of foxes is given by $$ N=\frac{37.5}{0.25+0.76^{t}} \text { foxes } $$ a. How many foxes were introduced into the protected area? b. Make a graph of \(N\) versus \(t\) and explain in words how the population of foxes increases with time. c. When will the fox population reach 100 individuals?
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