Math Problem Statement
Find the amount of an annuity with income function c(t), interest rate r, and term T. (Round your answer to two decimal places.) c(t) = $7000, r = 5%, T = 11 years
Solution
To find the amount of an annuity, we use the formula for the future value of a continuous annuity:
Here:
- (constant income function),
- (interest rate in decimal form),
- years (term).
Step 1: Set up the integral
Since is constant, we can simplify the integral:
Factor out the constant :
Step 2: Simplify the exponent
Rewrite as :
Since is constant:
Step 3: Solve the integral
The integral of is:
Evaluate this from to :
Substitute the limits:
Simplify:
Step 4: Calculate numerical values
Step 5: Final calculation
Substitute back into the formula for :
Calculate :
Final Answer:
Would you like details on any of these steps or further clarifications?
Related Questions:
- How is the future value of a continuous annuity different from that of a discrete annuity?
- Can this formula be applied if is not constant but a function of ?
- What are the practical implications of increasing or in annuities?
- How does compound interest relate to the annuity formula?
- What happens if the interest rate is compounded continuously instead of discretely?
Tip:
Always double-check your interest rate conversions (percentage to decimal) and rounding to ensure precise results in financial calculations.
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Math Problem Analysis
Mathematical Concepts
Calculus
Annuities
Exponential Growth
Integration
Formulas
A = ∫_0^T c(t) e^{r(T-t)} dt
A = 7000 e^{0.55} ∫_0^T e^{-0.05t} dt
Theorems
Exponential Growth
Integral Calculus
Continuous Compounding
Suitable Grade Level
College-Level
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