Math Problem Statement

Bryan plans to deposit $1500 each year into an annuity. If the account pays interest at a rate of 6.7 percent compounded continuously, find the approximate balance of his account after 7 years. Round any intermediate calculations to no less than six decimal places, and round your final answer to two decimal places.

Solution

The formula for the future value of an annuity with continuous compounding is given by:

A=Pert1er1A = P \cdot \frac{e^{rt} - 1}{e^r - 1}

Where:

  • AA is the balance of the account after tt years.
  • PP is the payment deposited each year ($1500).
  • rr is the annual interest rate (6.7% or 0.067).
  • tt is the number of years the money is deposited (7 years).
  • ee is the base of the natural logarithm (approximately 2.71828).

Plugging in the known values:

A=1500e0.06771e0.0671A = 1500 \cdot \frac{e^{0.067 \cdot 7} - 1}{e^{0.067} - 1}

Step 1: Calculate the terms inside the equation

First, calculate e0.0677e^{0.067 \cdot 7} and e0.067e^{0.067}.

  1. 0.0677=0.4690.067 \cdot 7 = 0.469
  2. e0.4691.598741e^{0.469} \approx 1.598741
  3. e0.0671.069e^{0.067} \approx 1.069

Step 2: Substitute these values into the formula

A=15001.59874111.0691A = 1500 \cdot \frac{1.598741 - 1}{1.069 - 1} A=15000.5987410.069A = 1500 \cdot \frac{0.598741}{0.069} A15008.6706A \approx 1500 \cdot 8.6706 A13005.93A \approx 13005.93

Final Answer:

The balance of Bryan's account after 7 years is approximately $13,005.93.

Let me know if you'd like more details or have any questions!


Follow-up Questions:

  1. What would the balance be if the interest rate were 5% instead of 6.7%?
  2. How does continuous compounding compare to monthly compounding in this case?
  3. What other factors might influence the actual balance of the account (e.g., fees or additional deposits)?
  4. How would the balance change if Bryan made deposits every six months instead of annually?
  5. What is the formula for the balance of an annuity with monthly compounding?

Tip: The more frequent the compounding, the higher the total accumulated interest over time. Continuous compounding maximizes this effect.

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Math Problem Analysis

Mathematical Concepts

Annuities
Continuous Compounding
Exponential Growth
Interest Calculations

Formulas

A = P * (e^(rt) - 1) / (e^r - 1)

Theorems

Future value of an annuity with continuous compounding

Suitable Grade Level

Grades 11-12