Chapter 5 - Exponential and Logarithmic Functions - 5.7 Financial Models - 5.7 Assess Your Understanding - Page 321: 42

$16.62\text{ years}$.

Work Step by Step

The formula for continuous compounding where $r$ is the rate of interest, $t$ is the time in years is, $P$ is the principal, $A$ is the amount you get back after $t$ years: $A=Pe^{rt}$. Here we have: $A=\$80000P=\$25000$ $r=7\%=0.07$. Substitute these values into the formula above to obtain: $\$80000=\$25000 \cdot e^{0.07t}\\\frac{$80000}{\$25000}=e^{0.07t}\\3.2=e^{0.07t}\\\ln{3.2}=\ln{e^{0.07t}}\\ln{3.2}=0.07t\\\frac{ln{3.2}}{0.07}=\frac{0.07t}{0.07}\\t=\frac{ln3.2}{0.07}=16.6164\approx16.62\text{ years}$.

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