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Chapter 2: Problem 12

If you borrow \(\$ 5000\) at an APR of \(r\) (as a decimal) from a lending institution that compounds interest continuously, and if you wish to pay off the note in 3 years, then your monthly payment \(M\), in dollars, can be calculated using $$ M=\frac{5000\left(e^{r / 12}-1\right)}{1-e^{-3 r}} $$ Your budget will allow a payment of \(\$ 150\) per month, and you are shopping for an interest rate that will give a payment of this size. What interest rate do you need to find?

Short Answer

Expert verified
An APR of approximately 3.2% is needed for a $150 monthly payment.

Step by step solution

01

Identify the Known Variables

From the problem statement, we know that the loan amount is \(5000, the desired monthly payment is \)150, and the loan term is 3 years. The unknown variable is the interest rate, \(r\), and our goal is to solve for it.
02

Set Up the Equation

Use the provided formula for monthly payment, \(M\). Substitute the known values into the equation: \(150 = \frac{5000(e^{r/12} - 1)}{1 - e^{-3r}}\).
03

Rearrange the Equation

To solve for \(r\), multiply both sides of the equation by \(1 - e^{-3r}\) to clear the denominator: \[150(1 - e^{-3r}) = 5000(e^{r/12} - 1).\]
04

Expand and Simplify

Expand the left side: \[150 - 150e^{-3r} = 5000e^{r/12} - 5000.\] Rearrange to group like terms: \[150 + 5000 = 5000e^{r/12} + 150e^{-3r}.\] This simplifies to: \[5150 = 5000e^{r/12} + 150e^{-3r}.\]
05

Solving for \(r\)

Solving for \(r\) directly in this form is complex and usually requires numerical methods or software to find an approximate value. However, there isn't a simple algebraic solution to find \(r\). We would typically use a numerical solver like a graphing calculator, specific mathematical software, or iterative methods such as the Newton-Raphson method to estimate \(r\).
06

Estimate \(r\) Using a Numerical Solver

Using a calculator or software for numerical solutions, input the equation \(5150 = 5000e^{r/12} + 150e^{-3r}\) and adjust \(r\) until the equation is satisfied. Through numerical methods, you will find that \(r \approx 0.032\) (or 3.2% as an APR).

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Key Concepts

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

APR (Annual Percentage Rate)
Understanding the Annual Percentage Rate (APR) is vital when dealing with loans or borrowing money. APR represents the annual cost of borrowing money, expressed as a percentage. This percentage includes not only the interest on the loan but also any other associated financial charges or fees.

Here's why APR is significant:
  • It provides a more comprehensive picture of the cost of borrowing compared to the nominal interest rate.
  • It helps you compare different loan offers effectively.
  • Generally, a lower APR indicates a more cost-effective loan.

In our given problem, the APR is a critical factor because it will determine what interest rate allows for a monthly payment of $150 on a $5000 loan over three years. Recognizing this helps in selecting the most favorable loan and managing repayment effectively.
Loan Payment Calculation
Calculating loan payments involves knowing variables such as the loan principal, the interest rate, and the term of the loan. For continuous compounding, the formula offers a unique way of determining payments compared to ordinary loans. A critical component of this calculation is the interest rate.

The formula given is:
\[M = \frac{5000(e^{r / 12}-1)}{1-e^{-3r}}\]

Here's what each component means:
  • \(M\) is the monthly payment.
  • The loan amount (principal) is $5000.
  • The factor \(e^{r/12}\) adjusts for monthly compounding.
  • The denominator \(1 - e^{-3r}\) considers the 3-year term in continuous compounding.

Through this calculation, you determine the monthly payment needed, ensuring that it fits within your budget constraints and the specific loan term.
Exponential Equations
Exponential equations often appear in financial mathematics, particularly in scenarios involving continuous compounding interest. Such equations typically involve an unknown variable in the exponent, as shown in our exercise.

For example, the expression:
\[150 = \frac{5000(e^{r/12} - 1)}{1 - e^{-3r}}\]
is a typical form of an exponential equation.

Key points about handling exponential equations:
  • Exponentials grow rapidly, making small changes in the exponent lead to large variations in the result.
  • Solving these equations often requires isolating the exponential term first.
  • Because they can't always be solved algebraically, numerical methods often become necessary.

This equation shows how compound interest factors into the calculations, requiring more advanced problem-solving techniques compared to simple interest.
Numerical Methods
Numerical methods are key when solving equations that do not have straightforward algebraic solutions. These methods allow us to approximate the root, or solution, to equations like the one in our loan exercise.

In the context of our problem, numerical solvers or iterative methods are essential for determining the unknown interest rate, \(r\). Here鈥檚 how these methods typically work:
  • They iteratively guess and check values for \(r\) in the equation \(5150 = 5000e^{r/12} + 150e^{-3r}\) until the left and right sides balance.
  • Software tools like graphing calculators, Python libraries, or Excel can automate this process, saving time and reducing errors.
  • Common techniques include the Newton-Raphson method, which adjusts guesses based on derivative calculations for faster convergence.

Numerical methods bridge the gap between theoretical equations and real-world applications, offering practical solutions where traditional algebra might fall short.

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Most popular questions from this chapter

Astronauts looking at Earth from a spacecraft can see only a portion of the surface. 10 See Figure \(2.59\) on the next page. The fraction \(F\) of the surface of Earth that is visible at a height \(h\), in kilometers, above the surface is given by the formula $$ F=\frac{0.5 h}{R+h} . $$ Here \(R\) is the radius of Earth, about 6380 kilometers. (For comparison, 1 kilometer is about \(0.62\) mile, and the moon is about 380,000 kilometers from Earth.) a. Make a graph of \(F\) versus \(h\) covering heights up to 100,000 kilometers. b. A value of \(F\) equal to \(0.25\) means that \(25 \%\), or one-quarter, of Earth's surface is visible. At what height is this fraction visible? c. During one flight of a space shuttle, astronauts performed an extravehicular activity at a height of 280 kilometers. What fraction of the surface of Earth is visible at that height? d. Is the graph of \(F\) concave up or concave down? Explain your answer in practical terms. e. Determine the limiting value for \(F\) as the height \(h\) gets larger. Explain your answer in practical terms.

The total number \(P\) of prey taken by a predator depends on the availability of prey. \(C\). \(S\). Holling proposed a function of the form \(P=c n /(1+d n)\) to model the number of prey taken in certain situations. \({ }^{25}\) Here \(n\) is the density of prey available, and \(c\) and \(d\) are constants that depend on the organisms involved as well as on other environmental features. Holling took data gathered earlier by T. Burnett on the number of sawfly cocoons found by a small wasp parasite at given host density. In one such experiment conducted, Holling found the relationship $$ P=\frac{21.96 n}{1+2.41 n} $$ where \(P\) is the number of cocoons parasitized and \(n\) is the density of cocoons available (measured as number per square inch). a. Draw a graph of \(P\) versus \(n\). Include values of \(n\) up to 2 cocoons per square inch. b. What density of cocoons will ensure that the wasp will find and parasitize 6 of them? c. There is a limit to the number of cocoons that the wasp is able to parasitize no matter how readily available the prey may be. What is this upper limit?

An enterprise rents out paddleboats for all-day use on a lake. The owner knows that he can rent out all 27 of his paddleboats if he charges \(\$ 1\) for each rental. He also knows that he can rent out only 26 if he charges \(\$ 2\) for each rental and that, in general, there will be 1 less paddleboat rental for each extra dollar he charges per rental. a. What would the owner's total revenue be if he charged \(\$ 3\) for each paddleboat rental? b. Use a formula to express the number of rentals as a function of the amount charged for each rental. c. Use a formula to express the total revenue as a function of the amount charged for each rental. d. How much should the owner charge to get the largest total revenue?

The manager of an employee health plan for a firm has studied the balance \(B\), in millions of dollars, in the plan account as a function of \(t\), the number of years since the plan was instituted. He has determined that the account balance is given by the formula \(B=60+7 t-50 e^{0.1 t}\). a. Make a graph of \(B\) versus \(t\) over the first 7 years of the plan. b. At what time is the account balance at its maximum? c. What is the smallest value of the account balance over the first 7 years of the plan?

You own a motel with 30 rooms and have a pricing structure that encourages rentals of rooms in groups. One room rents for \(\$ 85\), two rent for \(\$ 83\) each, and in general the group rate per room is found by taking \(\$ 2\) off the base of \(\$ 85\) for each extra room rented. a. How much money do you take in if a family rents two rooms? b. Use a formula to give the rate you charge for each room if you rent \(n\) rooms to an organization. c. Find a formula for a function \(R=R(n)\) that gives the revenue from renting \(n\) rooms to a convention host. d. What is the most money you can make from rental to a single group? How many rooms do you rent?
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