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 of an investment compounded continuously, we use the formula for continuous compounding:
Where:
- is the principal amount (initial investment),
- is the annual interest rate (in decimal form),
- is the time the money is invested for (in years),
- is Euler's number (approximately 2.71828).
Given:
- dollars,
- ,
- years.
Now, let's compute the amount :
I'll calculate this value.The amount 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:
- What would the amount be if the interest rate were 10% instead of 9%?
- How would the total change if the investment period were 40 years?
- What is the formula for compounding interest monthly or annually instead of continuously?
- How does continuous compounding differ from other types of compounding?
- 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
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