cooper wants to start a business, which he estimates will cost $7,248.00. if cooper puts $2,000.00 into a…

Explanation:

Step1: Recall continuous - compounding formula

The formula for continuous - compounding is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), and $t$ is the time in years. We know that $A=$7248$, $P = $2000$, and $r=0.14$. Substitute these values into the formula: $7248 = 2000e^{0.14t}$.

Step2: Solve for $t$

First, divide both sides of the equation by 2000: $\frac{7248}{2000}=e^{0.14t}$. So, $3.624 = e^{0.14t}$. Then, take the natural logarithm of both sides: $\ln(3.624)=\ln(e^{0.14t})$. Since $\ln(e^{x}) = x$, we have $\ln(3.624)=0.14t$. Now, solve for $t$: $t=\frac{\ln(3.624)}{0.14}$. We know that $\ln(3.624)\approx1.285$ and $t=\frac{1.285}{0.14}\approx9.179$ years.

Step3: Convert years to years and months

The whole - number part of $t$ is the number of years, which is 9 years. The decimal part $0.179$ of a year is converted to months. Since there are 12 months in a year, the number of months is $0.179\times12\approx2.15$. Rounding to the nearest month, we get 2 months.

Answer:

9 years and 2 months