Question

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?

Answer 1

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:
`\displaystyle A = Pe^(rt)`

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

Compounded Annually Interest Rate Formula:
`\displaystyle A = P(1 + r)^t`

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

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]:
   `\displaystyle 2P = 5500e^\big{0.085t}`
2. Substitute in P:
   `\displaystyle 2(5500) = 5500e^\big{0.085t}`
3. Simplify:
   `\displaystyle 2 = e^\big{0.085t}`
4. Isolate t term:
   `\displaystyle \ln 2 = 0.085t`
5. Isolate t:
   `\displaystyle t = (\ln 2)/(0.085)`

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]:
   `\displaystyle A = 5500(1 + 0.0825)^\big{(\ln 2)/(0.085)}`
2. Evaluate:
   `\displaystyle A = 10498.10`

∴ 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