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:

FV=RerT1rFV = R \cdot \frac{e^{rT} - 1}{r}

Where:

  • RR is the annual payment rate (2000 dollars per year),
  • rr is the interest rate as a decimal (0.0180.018),
  • TT is the time in years (from 12 to 18, so 1812=618 - 12 = 6),
  • ee is the base of the natural logarithm.

Plug in the values:

FV=2000e0.018610.018FV = 2000 \cdot \frac{e^{0.018 \cdot 6} - 1}{0.018}

Step-by-step calculation:

  1. Compute 0.0186=0.1080.018 \cdot 6 = 0.108.
  2. Compute e0.1081.114141e^{0.108} \approx 1.114141.
  3. Subtract 1: 1.1141411=0.1141411.114141 - 1 = 0.114141.
  4. Divide by 0.0180.018:
    0.1141410.0186.34117\frac{0.114141}{0.018} \approx 6.34117
  5. Multiply by 20002000:
    20006.34117126822000 \cdot 6.34117 \approx 12682

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:

  1. What if the interest rate was 2% instead of 1.8%?
  2. How does the future value change if the contributions stopped at age 16?
  3. What happens if the interest compounds quarterly instead of continuously?
  4. How do we modify the formula for irregular contributions?
  5. 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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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