Question

how much should be invested now at an interest rate of 5.5% per year, compounded continuously, to have $20000 in two years? do not round any intermediate computations, and round your answer to the nearest cent.

Explanation:

Step1: Recall continuous compounding formula

The formula for continuous compounding is \( A = Pe^{rt} \), where \( A \) is the amount after time \( t \), \( P \) is the principal (initial investment), \( r \) is the annual interest rate (in decimal), and \( t \) is the time in years. We need to solve for \( P \), so rearrange the formula to \( P=\frac{A}{e^{rt}} \).

Step2: Identify values

Given \( A = 20000 \), \( r = 5.5\%=0.055 \), and \( t = 2 \) years.

Step3: Substitute values into the formula

First, calculate \( rt = 0.055\times2 = 0.11 \). Then, \( e^{0.11}\approx1.116278 \) (using the exponential function). Now, \( P=\frac{20000}{e^{0.11}}=\frac{20000}{1.116278}\approx17917.43 \).

Answer:

\(\$17917.43\)