Respuesta :
To answer this question, we have to use the next formula, which is the continuous compounding interest formula:
[tex]A=Pe^{rt}_{}[/tex]Where:
• A is the accrued amount. In this case, we have A = $5,200.
,• P is the principal. This is the amount we need to find.
,• r is the interest rate. In this case, the value is r = 7.91% or 7.91/100.
,• t is time in years. In this case, t = 10 years.
,• e is the "e" number (e is approximately 2.71828182846)
Therefore, we have:
[tex]\begin{gathered} A=Pe^{rt}_{} \\ 5200=Pe^{\frac{7.91}{100}\cdot10} \end{gathered}[/tex]Now, we have to solve the equation by using the inverse function of the natural exponential function, namely, the natural logarithm as follows:
[tex]\begin{gathered} 5200=Pe^{7.91\cdot\frac{10}{100}} \\ 5200=Pe^{7.91\cdot\frac{1}{10}} \\ 5200=Pe^{\frac{7.91}{10}}=Pe^{0.791} \\ \end{gathered}[/tex]Now, we can see that is not necessary to use the natural logarithm. We have to divide both sides by the resulting exponential value:
[tex]\begin{gathered} \frac{5200}{e^{0.791}}=P\frac{e^{0.791}}{e^{0.791}}\Rightarrow\frac{e^{0.791}}{e^{0.791}}=1,\frac{a}{a}=1 \\ \end{gathered}[/tex]Therefore, we finally have:
[tex]P=\frac{5200}{e^{0.791}}=2357.63412214[/tex]If we round the resulting value to two decimal places, we have that:
[tex]P=\$2357.63[/tex]