shivani earned some money doing odd jobs last summer, but wants to buy a bike that costs twice that much. if…

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 want to find the time $t$ when $A = 2P$ and $r=0.15$. Substitute $A = 2P$ and $r = 0.15$ into the formula: $2P=Pe^{0.15t}$.

Step2: Solve for $t$

Divide both sides of the equation $2P = Pe^{0.15t}$ by $P$ (since $P\neq0$). We get $2=e^{0.15t}$. Take the natural logarithm of both sides: $\ln(2)=\ln(e^{0.15t})$. Since $\ln(e^{x}) = x$, the equation simplifies to $\ln(2)=0.15t$. Then, solve for $t$: $t=\frac{\ln(2)}{0.15}$. We know that $\ln(2)\approx0.6931$, so $t=\frac{0.6931}{0.15}\approx4.62$ years.

Step3: Convert years to years and months

The whole - number part of $t$ gives the number of years, which is $4$ years. The decimal part $0.62$ of a year is converted to months. Since there are $12$ months in a year, the number of months is $0.62\times12 = 7.44\approx7$ months.

Answer:

4 years and 7 months