To solve this problem, we start with the formula for continuous compounding interest, which is given by:
[tex]\[ A = P \cdot e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount to which the investment grows,
- [tex]\( P \)[/tex] is the principal or initial investment,
- [tex]\( r \)[/tex] is the interest rate per year (expressed as a decimal),
- [tex]\( t \)[/tex] is the time in years,
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
For the given problem:
- The initial investment [tex]\( P \)[/tex] is [tex]$3000,
- The time \( t \) is 13 years,
- The interest rate \( r \) needs to be expressed as a decimal (e.g., 1% = 0.01).
Using the formula \( A = 3000 \cdot e^{13r} \), we can calculate the amounts for various interest rates \( r \).
Let's fill in the amounts for each of the given interest rates:
1. For an interest rate of 1% (0.01):
\[
A = 3000 \cdot e^{13 \times 0.01} = 3000 \cdot e^{0.13}
\]
This computation gives an amount of \( \$[/tex]3416.49 \).
2. For an interest rate of 2% (0.02):
[tex]\[
A = 3000 \cdot e^{13 \times 0.02} = 3000 \cdot e^{0.26}
\][/tex]
This computation gives an amount of [tex]\( \$3890.79 \)[/tex].
3. For an interest rate of 3% (0.03):
[tex]\[
A = 3000 \cdot e^{13 \times 0.03} = 3000 \cdot e^{0.39}
\][/tex]
This computation gives an amount of [tex]\( \$4430.94 \)[/tex].
4. For an interest rate of 4% (0.04):
[tex]\[
A = 3000 \cdot e^{13 \times 0.04} = 3000 \cdot e^{0.52}
\][/tex]
This computation gives an amount of [tex]\( \$5046.08 \)[/tex].
5. For an interest rate of 5% (0.05):
[tex]\[
A = 3000 \cdot e^{13 \times 0.05} = 3000 \cdot e^{0.65}
\][/tex]
This computation gives an amount of [tex]\( \$5746.62 \)[/tex].
6. For an interest rate of 6% (0.06):
[tex]\[
A = 3000 \cdot e^{13 \times 0.06} = 3000 \cdot e^{0.78}
\][/tex]
This computation gives an amount of [tex]\( \$6544.42 \)[/tex].
So the completed table should look like this:
[tex]\[
\begin{tabular}{|c|c|}
\hline \begin{tabular}{l}
Rate \\
per \\
Year
\end{tabular} & Amount \\
\hline 1\% & \$3416.49 \\
\hline 2\% & \$3890.79 \\
\hline 3\% & \$4430.94 \\
\hline 4\% & \$5046.08 \\
\hline 5\% & \$5746.62 \\
\hline 6\% & \$6544.42 \\
\hline
\end{tabular}
\][/tex]
