Math Problem Statement
How many years will it take $ 6 comma 000$6,000 to grow to $ 11 comma 600$11,600 if it is invested at 3.75 %3.75% compounded continuously? Question content area bottom Part 1 enter your response here years (Round to two decimal places.)
Solution
To solve this, we will use the formula for continuous compounding, which is given by:
Where:
- is the final amount (11,600),
- is the initial principal (6,000),
- is the annual interest rate (3.75% or 0.0375),
- is the time in years,
- is the base of the natural logarithm.
We need to solve for . First, rearrange the formula to isolate :
Now, substitute the given values into the equation:
Let's calculate that.
It will take approximately 17.58 years for $6,000 to grow to $11,600 if it is invested at an interest rate of 3.75% compounded continuously.
Would you like more details or have any questions?
Here are 5 related questions:
- What if the interest rate was 5%? How long would it take?
- How much would $6,000 grow to in 10 years at 3.75% continuous compounding?
- What is the difference between continuous and annual compounding in terms of growth?
- Can you derive the continuous compounding formula?
- How does changing the principal amount affect the growth period?
Tip: Continuous compounding allows money to grow at the fastest rate possible for a given interest rate.
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Math Problem Analysis
Mathematical Concepts
Exponential Growth
Continuous Compounding
Natural Logarithms
Formulas
A = P * e^(rt)
t = ln(A/P) / r
Theorems
Exponential Growth Theorem
Logarithmic Properties
Suitable Grade Level
Grades 10-12
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