Math Problem Statement
Solution
The problem asks to use the continuous compound interest formula to find the time when given the final amount , principal , and interest rate .
The formula for continuous compound interest is:
where:
- is the final amount,
- is the initial principal,
- is the interest rate (as a decimal),
- is the time in years,
- is the base of the natural logarithm (approximately 2.71828).
We are given:
- ,
- ,
- .
Now, we need to solve for . Follow these steps:
Step 1: Plug in the values
Step 2: Divide both sides by
Step 3: Take the natural logarithm of both sides
Step 4: Solve for
Thus, years.
Would you like more details on any of the steps?
Here are 5 related questions to explore:
- How does continuous compound interest differ from regular compound interest?
- Can you solve for if is known?
- How would the formula change if interest were compounded monthly instead of continuously?
- What is the effect of doubling the principal on the time ?
- How does the value of influence the behavior of the equation?
Tip: Always remember to convert percentage rates to decimals (e.g., 9.1% = 0.091) before using them in formulas.
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Continuous Compound Interest
Natural Logarithms
Formulas
Continuous Compound Interest Formula: A = P * e^(rt)
Natural Logarithm: ln(x)
Theorems
Properties of Exponential Functions
Logarithmic Inversion for Exponential Equations
Suitable Grade Level
Grades 10-12