/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 APPLICATION A credit card accoun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 8

APPLICATION A credit card account is essentially a loan. A constant percent interest is added to the balance. Stanley buys \(\$ 100\) worth of groceries with his credit card. The balance then grows by \(1.75 \%\) interest each month. How much will he owe if he makes no payments in 4 months? Write the expression you used to do this calculation in expanded form and also in exponential form. (a)

Short Answer

Expert verified
Stanley will owe approximately $107.18 after 4 months.

Step by step solution

01

Understanding the Initial Conditions

Stanley has an initial debt of $100 on his credit card. This amount will grow by a constant monthly interest rate of 1.75% if no payments are made. Our task is to find the balance after 4 months with compounding interest.
02

Convert the Interest Rate to Decimal Form

The monthly interest rate provided is 1.75%. To use this in calculations, convert it to decimal form by dividing by 100: \( 1.75\% = \frac{1.75}{100} = 0.0175 \).
03

Set Up the Expression in Expanded Form

The expanded form shows the balance compounding month by month. After 1 month, the balance is \( 100 \times (1 + 0.0175) \) .After 4 months, the expression in expanded form is:\[100 \times (1 + 0.0175) \times (1 + 0.0175) \times (1 + 0.0175) \times (1 + 0.0175)\]
04

Simplify to Exponential Form

The expanded form can be represented in exponential form due to repeated multiplication:\[100 \times (1 + 0.0175)^4\]This shows the initial amount compounded over 4 months.
05

Calculate the Final Amount

Now, calculate the amount Stanley owes by evaluating the exponential expression:\[100 \times (1.0175)^4 \approx 100 \times 1.0718 = 107.18\]The final amount Stanley owes is approximately $107.18.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Exponential Growth
Exponential growth often appears in the context of compound interest, where an amount increases by a certain percentage over regular intervals. In Stanley's case, his credit card balance grows exponentially because it is increased by the same percentage each month. Each time the balance grows, it does so not just on the initial amount, but also on the previous month's increase.

This can be visualized with the mathematical expression for exponential growth: \[ A = P(1 + r)^t \]where:
  • A is the amount of money accumulated after a certain number of intervals (months in this case).
  • P is the principal amount or the initial amount of money (in this scenario, $100).
  • r is the rate of interest (0.0175 here).
  • t is the time the money is invested or borrowed for in months (4 months for Stanley).
This formula shows how the balance compounds over multiple periods, thus demonstrating exponential growth.
Financial Arithmetic
Financial arithmetic involves calculating various financial scenarios using mathematical operations. In the case of Stanley and his credit card debt, it's about understanding how his balance increases over time due to interest.

To better understand this, consider how each month's interest is calculated. For Stanley:
  • During the first month, the interest is calculated on the initial $100, growing the balance to \(100 \times (1 + 0.0175)\).
  • In the second month, the interest calculation uses the new balance from the first month, continuing as \((100 \times (1.0175)) \times (1.0175)\).
This pattern repeats for each subsequent month. Financial arithmetic relies heavily on these iterative calculations to predict outcomes over time, ensuring one understands how values change due to interest rates and time lapses.
Interest Calculation
Interest calculation is a fundamental part of financial transactions, notably in loans and savings. When using compound interest, as Stanley's case illustrates, it involves calculating interest on both the initial principal and accrued interest from previous periods.

For simple calculations, convert the percentage rate to a decimal; Stanley's rate of 1.75% becomes 0.0175. Then, use this decimal in formulas to evaluate the compounded amount:

Stanley's debt calculation derived from:
  • Expanded Form: Reflects every individual increment: \(100 \times (1 + 0.0175) \times (1 + 0.0175) \times (1 + 0.0175) \times (1 + 0.0175)\).
  • Exponential Form: Summarizes these increments succinctly: \(100 \times (1.0175)^4\).
  • Final calculation leads to an approximate debt of $107.18 after four months.
Understanding interest calculations helps in making informed financial decisions, distinguishing between simple and compound interest scenarios, and planning effectively for future financial standings.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

APPLICATION In the course of a mammal's lifetime, \(i\) heart beats about 800 million times, regardless of the mammal's size or weight. (This excludes humans.) a. An elephant's heart beats approximately 25 times a minute. How many years would you expect an elephant to live? Use scientific notation to calculate your answer. (a) b. A pygmy shrew's heart beats approximately 1150 times a minute. How many years would you expect a pygmy shrew to live? c. If this relationship were true for humans, how many years would you expect a human being with a heart rate of 60 bpm to live?

The equation \(y=262(1+0.0593)^{x}\) models the frequency in hertz of various notes on the piano, with middle \(\mathrm{C}\) considered as note 0 . The average human ear can detect frequencies between 20 and 20,000 hertz If a piano keyboard were extended, the nighest and lowest notes audible to the average human ear would be how far above and below middle \(\mathrm{C}\) ? (it)

Use the properties of exponents to rewrite each expression. Your final answer should have only positive exponents. Use calculator tables to check that your expression is equivalent to the original expression. a. \(\frac{x \cdot x \cdot x}{x}\) b. \(2 x^{-1}\) c. \(\frac{6.273 x^{8}}{5.1 x^{3}}\) d. \(3^{-x}\) e. \(3 x^{0}\) f. \(x^{2} \cdot x^{5}\) g. \(\left(3^{4}\right)^{x}\) h. \(\frac{1}{x^{-2}}\)

Match expressions from this list that are equivalent but written in different exponential forms. There can be multiple matches. a. \(\left(4 x^{4}\right)(3 x)\) b. \(\left(8 x^{2}\right)\left(3 x^{2}\right)\) c. \((12 x)(4 x)\) d. \(\left(6 x^{3}\right)\left(2 x^{2}\right)\) e. \(12 x^{6}\) f. \(24 x^{4}\) g. \(12 x^{5}\) h. \(48 x^{2}\)

There are many stories in children's literature that involve magic pots. An Italian variation goes something like this: A woman puts a pot of water on the stove to boil. She says some special words, and the pot begins filling with pasta. Then she says another set of special words, and the pot stops filling up. Suppose someone overhears the first words, takes the pot, and starts it in its pasta-creating mode. Two liters of pasta are created. Then the pot continues to create more pasta because the impostor doesn't know the second set of words. The volume continues to increase \(50 \%\) per minute. a. Write an equation that models the amount of pasta in liters, \(y\), after \(x\) minutes. (a) b. How much pasta will there be after 30 seconds? c. How much pasta will there be after 10 minutes? d. How long, to the nearest second, will it be until the entire house, which can hold 450,000 liters, is full of pasta?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.