Math Problem Statement

Jace invested $380 in an account paying an interest rate of 6.2% compounded continuously. Assuming no deposits or withdrawals are made, how much money, to the nearest ten dollars, would be in the account after 15 years?

Solution

To calculate how much money Jace will have in the account after 15 years with continuous compounding interest, we use the formula for continuous compound interest:

A=PertA = P \cdot e^{rt}

Where:

  • AA is the amount of money in the account after time tt,
  • PP is the principal (initial investment),
  • rr is the annual interest rate (expressed as a decimal),
  • tt is the time in years,
  • ee is Euler's number (approximately 2.71828).

Given values:

  • P=380P = 380
  • r=6.2%=0.062r = 6.2\% = 0.062
  • t=15t = 15

Now, plug the values into the formula:

A=380e0.06215A = 380 \cdot e^{0.062 \cdot 15}

First, calculate the exponent:

0.06215=0.930.062 \cdot 15 = 0.93

Now, calculate e0.93e^{0.93}:

e0.932.529e^{0.93} \approx 2.529

Next, multiply this by the principal:

A=3802.529962.02A = 380 \cdot 2.529 \approx 962.02

Rounding to the nearest ten dollars:

A960A \approx 960

So, after 15 years, Jace would have $960 in the account.

Would you like to go over any steps in more detail, or do you have any other questions about this topic?

Here are some related questions you might find interesting:

  1. How would the result change if the interest were compounded annually instead of continuously?
  2. What is the formula for compound interest when compounded annually or monthly?
  3. How would the amount in the account change if the interest rate were 7% instead of 6.2%?
  4. How would you calculate the time it takes for an investment to double with continuous compounding?
  5. How does the principal amount affect the final value when using continuous compounding?

Tip: For continuous compounding, the rate of growth is always slightly higher than with other compounding methods because interest is added infinitely often.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Continuous Compounding
Exponential Growth
Compound Interest

Formulas

A = P * e^(rt)

Theorems

Exponential Growth with Continuous Compounding

Suitable Grade Level

Grades 9-12