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Zoe invested $5,500 in an account paying an interest rate of 8 1/2% compounded continuously. Aubrey invested $5,500 in an account paying an interest rate of 8 1/4% compounded annually. To the nearest dollar, how much money would Aubrey have in her account when Zoe's money has doubled in value?

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Answer:

Audrey will have $10,498.10 when Zoe's money has doubled in value.

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

Algebra I

Equality Properties

  • Multiplication Property of Equality
  • Division Property of Equality
  • Addition Property of Equality
  • Subtraction Property of Equality

Algebra II

Natural Logs and Exponentials

Compounded Continuously Interest Rate Formula: [tex]\displaystyle A = Pe^{rt}[/tex]

  • P is the principle amount
  • r is the interest rate
  • t is time

Compounded Annually Interest Rate Formula: [tex]\displaystyle A = P(1 + r)^t[/tex]

  • P is the principle amount
  • r is the interest rate
  • t is time

Step-by-step explanation:

Step 1: Define

Identify.

Zoe:

P = $5,500

r = 0.085

A = 2P

Aubrey:

P = $5,500

r = 0.0825

Step 2: Find Time

Find time elapsed by using Zoe.

  1. Substitute in variables [Compounded Continuously Interest Rate]:        [tex]\displaystyle 2P = 5500e^\big{0.085t}[/tex]
  2. Substitute in P:                                                                                             [tex]\displaystyle 2(5500) = 5500e^\big{0.085t}[/tex]
  3. Simplify:                                                                                                        [tex]\displaystyle 2 = e^\big{0.085t}[/tex]
  4. Isolate t term:                                                                                               [tex]\displaystyle \ln 2 = 0.085t[/tex]
  5. Isolate t:                                                                                                        [tex]\displaystyle t = \frac{\ln 2}{0.085}[/tex]

So the time it takes for Zoe to get double her money is approximately 8.15467 years.

Step 3: Find Audrey's Money

  1. Substitute in variables [Compounded Annually Interest Rate]:                [tex]\displaystyle A = 5500(1 + 0.0825)^\big{\frac{\ln 2}{0.085}}[/tex]
  2. Evaluate:                                                                                                       [tex]\displaystyle A = 10498.10[/tex]

∴ after the elapsed time of approximately 8.15467 years, Zoe would have made double her money valued at $11,000 and Audrey would have made $10,498.10.

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Topic: Algebra II

Unit: Logarithmic Functions

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