Math Problem Statement

Find the amount A in dollars of $8,000 invested for 30 years at 9% compounded continuously. (Enter a number. Round answer to the nearest cent.) $

Solution

To find the future amount AA of an investment compounded continuously, we use the formula for continuous compounding:

A=PertA = P e^{rt}

Where:

  • PP is the principal amount (initial investment),
  • rr is the annual interest rate (in decimal form),
  • tt is the time the money is invested for (in years),
  • ee is Euler's number (approximately 2.71828).

Given:

  • P=8000P = 8000 dollars,
  • r=9%=0.09r = 9\% = 0.09,
  • t=30t = 30 years.

Now, let's compute the amount AA:

A=8000×e(0.09×30)A = 8000 \times e^{(0.09 \times 30)}

I'll calculate this value.The amount AA after 30 years of continuous compounding at 9% interest is $119,037.85.

Do you have any further questions or would you like a detailed explanation of the steps?

Here are five related questions to consider:

  1. What would the amount be if the interest rate were 10% instead of 9%?
  2. How would the total change if the investment period were 40 years?
  3. What is the formula for compounding interest monthly or annually instead of continuously?
  4. How does continuous compounding differ from other types of compounding?
  5. What would be the effect on the amount if the initial investment was $10,000 instead of $8,000?

Tip: Continuous compounding grows faster than other compounding methods because it calculates interest at an infinitely small interval.

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Math Problem Analysis

Mathematical Concepts

Continuous Compounding
Exponential Growth
Interest Calculation

Formulas

A = P e^{rt}

Theorems

Exponential Growth Theorem

Suitable Grade Level

Grades 11-12