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5.6: Corequisite- Exponential Models (Practice and Application)

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    SPECIFIC OBJECTIVES

    By the end of this lesson, you should understand that

    • solving for a variable includes isolating it by “undoing” the actions to it.
    • there are differences and similarities between exponential growth and decay.

    By the end of this lesson, you should be able to

    • explicitly write out the order of operations to evaluate a given equation.
    • use the compound interest formula for different compounding periods.
    • write an exponential decay model.

    INTRODUCTION

    We spent some time at the start of this module discussing how to “undo” operations to solve linear equations. We will also need to “undo” operations to solve the more complicated exponential equations. Let’s review our more familiar linear situation.

    (1) Consider the following equation for the values given in (a) and (b). In each case, write out the steps (using words) that you would take to solve for the variable. Remember that we want to apply this process to exponential equations soon. So, think carefully about what you do in each step while solving.

    \(y = −4x – 2\)

    (a) Solve for y if x = −3. Using words, explain your steps.

    (b) Solve for x if y = −3. Using words, explain your steps.

    PROBLEM SITUATION 1: THE VALUE OF A CD

    In the previous corequisite lesson, you used the following formula to learn how to find the accumulated amount (A) of a certificate of deposit (CD) with a compounding period of one year:

    \(A = P(1 + r)^t\)

    In this problem situation, you will use the same formula to explore the accumulated amount (A) of a CD with a shorter compounding period. Remember that the variable (P) represents the principal, r represents the interest rate per period, and t represents the number of periods.

    (2) (a) Suppose you invest $1,000 in a two-year CD, advertised with an annual percentage rate (APR) of 12%, where compounding occurs monthly. If the APR is 12% per year, how much is it per month?

    (b) Use your answer to complete the following table. Record how you found the results in the middle column.

    • Principal = $1,000
    • APR = 12%
    • Term = two years
    • Compounding period = one month
    Period Calculation Amount Accrued
    1 month    
    2 months    
    3 months    
    6 months    
    12 months    
    24 months    

    (3) Write a general formula that can be used to calculate the value of any CD with any number of compounding periods in a year. Define your variables. Hint: Use the simplified compound formula as a starting point, but recognize the key difference in this current example: instead of interest being compounded once annually, the interest in this example is compounded monthly; that is, it is compounded 12 times a year!

    (4) How does the compounding period affect the accumulated amount? Hint: Use quarterly compounding to compute the accumulated amount after eight quarters (24 months), and compare to the results in the above table.

    PROBLEM SITUATION 2: UNDERSTANDING DEPRECIATION

    Depreciation is a process of losing value. For example, new automobiles lose 15% to 20% of their value each year for the first few years you own them.

    (5) (a) Based on this fact, develop a depreciated-value (D) formula for a $26,000 automobile. Assume that the depreciation is 15% per year. Use the table below to guide your calculations.

    Age of the Automobile Calculation Value
    New N/A $26,000
    1 year old    
    2 years old    
    5 years old    
    t years old    

    (b) Use your formula from part (a) to make a graph of the value of the vehicle over time. Note: If completing this problem online, follow the instructions given online to create your graph.


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