Math Problem Statement
When Pilar is
12
, her grandparents begin placing a continuous stream of
$
2
,
000
per year into a college fund which has a continuously compounding interest rate of
1.8
%
. What will be the value of this continuous stream when Pilar is
18
?
Round your answer to the nearest integer. Do not include a dollar sign or commas in your answer.
Solution
To determine the value of a continuous stream with continuously compounding interest, we use the formula for the future value of a continuous income stream:
Where:
- is the annual payment rate (2000 dollars per year),
- is the interest rate as a decimal (),
- is the time in years (from 12 to 18, so ),
- is the base of the natural logarithm.
Plug in the values:
Step-by-step calculation:
- Compute .
- Compute .
- Subtract 1: .
- Divide by :
- Multiply by :
Final Answer:
The value of the continuous stream when Pilar is 18 is 12682.
Would you like additional details on the steps or any related examples?
Related Questions:
- What if the interest rate was 2% instead of 1.8%?
- How does the future value change if the contributions stopped at age 16?
- What happens if the interest compounds quarterly instead of continuously?
- How do we modify the formula for irregular contributions?
- How would the value change if Pilar starts saving earlier at age 10?
Tip: Continuous compounding grows slightly faster than other methods due to its infinite compounding periods per year.
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Math Problem Analysis
Mathematical Concepts
Continuous Compound Interest
Exponential Growth
Future Value of a Continuous Stream
Formulas
FV = R * (e^(rT) - 1) / r
Theorems
Continuous compounding theorem
Suitable Grade Level
Grades 11-12
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