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:

A=0Tc(t)er(Tt)dtA = \int_0^T c(t) e^{r(T-t)} dt

Here:

  • c(t)=7000c(t) = 7000 (constant income function),
  • r=0.05r = 0.05 (interest rate in decimal form),
  • T=11T = 11 years (term).

Step 1: Set up the integral

Since c(t)c(t) is constant, we can simplify the integral:

A=0117000e0.05(11t)dtA = \int_0^{11} 7000 e^{0.05(11-t)} dt

Factor out the constant 70007000:

A=7000011e0.05(11t)dtA = 7000 \int_0^{11} e^{0.05(11-t)} dt

Step 2: Simplify the exponent

Rewrite 0.05(11t)0.05(11 - t) as 0.550.05t0.55 - 0.05t:

A=7000011e0.55e0.05tdtA = 7000 \int_0^{11} e^{0.55} e^{-0.05t} dt

Since e0.55e^{0.55} is constant:

A=7000e0.55011e0.05tdtA = 7000 e^{0.55} \int_0^{11} e^{-0.05t} dt

Step 3: Solve the integral

The integral of e0.05te^{-0.05t} is:

e0.05tdt=e0.05t0.05=20e0.05t\int e^{-0.05t} dt = \frac{e^{-0.05t}}{-0.05} = -20 e^{-0.05t}

Evaluate this from t=0t = 0 to t=11t = 11:

011e0.05tdt=20e0.05t011\int_0^{11} e^{-0.05t} dt = -20 e^{-0.05t} \Big|_0^{11}

Substitute the limits:

=20(e0.05(11)e0.05(0))= -20 \left( e^{-0.05(11)} - e^{-0.05(0)} \right)

Simplify:

e0.05(11)=e0.55ande0.05(0)=e0=1e^{-0.05(11)} = e^{-0.55} \quad \text{and} \quad e^{-0.05(0)} = e^{0} = 1

011e0.05tdt=20(e0.551)\int_0^{11} e^{-0.05t} dt = -20 \left( e^{-0.55} - 1 \right)

Step 4: Calculate numerical values

  • e0.550.57695e^{-0.55} \approx 0.57695

011e0.05tdt=20(0.576951)=20(0.42305)=8.461\int_0^{11} e^{-0.05t} dt = -20 \left( 0.57695 - 1 \right) = -20 (-0.42305) = 8.461

Step 5: Final calculation

Substitute back into the formula for AA:

A=7000e0.55×8.461A = 7000 e^{0.55} \times 8.461

Calculate e0.55e^{0.55}:

e0.551.73325e^{0.55} \approx 1.73325

A=7000×1.73325×8.461102,794.80A = 7000 \times 1.73325 \times 8.461 \approx 102,794.80

Final Answer:

102,794.80\boxed{102,794.80}

Would you like details on any of these steps or further clarifications?

Related Questions:

  1. How is the future value of a continuous annuity different from that of a discrete annuity?
  2. Can this formula be applied if c(t)c(t) is not constant but a function of tt?
  3. What are the practical implications of increasing TT or rr in annuities?
  4. How does compound interest relate to the annuity formula?
  5. 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