Math Problem Statement

Moneysaver's Bank offers a savings account that earns 5% interest per year, compounded continuously. If Sam deposits $1500, how much will he have in the account after four years, assuming he makes no withdrawals?

Solution

To calculate the amount Sam will have in his account after four years, we use the formula for continuously compounded interest:

A=PertA = P e^{rt}

Where:

  • AA is the amount in the account after time tt.
  • PP is the principal (initial deposit).
  • 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:

  • P=1500P = 1500 (initial deposit).
  • r=5%=0.05r = 5\% = 0.05 (annual interest rate).
  • t=4t = 4 years.

Now, plug in the values:

A=1500×e0.05×4A = 1500 \times e^{0.05 \times 4}

A=1500×e0.20A = 1500 \times e^{0.20}

We can now calculate the value of e0.20e^{0.20}.

A1500×1.2214A \approx 1500 \times 1.2214

A1832.10A \approx 1832.10

Result:

Sam will have approximately $1832.10 in the account after four years.

Would you like a more detailed explanation or have any further questions?

Related Questions:

  1. How would the amount change if the interest rate was 6% instead of 5%?
  2. What is the difference between continuous compounding and annual compounding?
  3. How long would it take for Sam’s deposit to double with continuous compounding at 5% interest?
  4. How much more would Sam earn after 4 years if the interest was compounded monthly instead of continuously?
  5. How would the account balance change if Sam made an additional deposit of $500 after two years?

Tip: When using continuous compounding, the formula A=PertA = P e^{rt} provides the most accurate result. However, for less frequent compounding, you can use the formula for compound interest with the number of periods per year.

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Math Problem Analysis

Mathematical Concepts

Continuous Compounding
Exponential Growth
Interest Calculation

Formulas

A = P e^(rt)

Theorems

Continuous compounding theorem

Suitable Grade Level

Grades 9-12