9. a principal amount of $4200, earns 3.6% interest compounded quarterly. how long does it take for the…

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. Given $P = 4200$, $r=0.036$, $n = 4$ (compounded quarterly), and $A = 15000$. Substitute these values into the formula: $15000=4200(1 +\frac{0.036}{4})^{4t}$

Step2: Simplify the equation

First, simplify $1+\frac{0.036}{4}=1 + 0.009=1.009$. The equation becomes $\frac{15000}{4200}=(1.009)^{4t}$. $\frac{25}{7}=(1.009)^{4t}$. Take the natural logarithm of both sides: $\ln(\frac{25}{7})=\ln((1.009)^{4t})$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(\frac{25}{7}) = 4t\ln(1.009)$.

Step3: Solve for $t$

We know that $\ln(\frac{25}{7})\approx\ln(25)-\ln(7)\approx3.218876 - 1.94591 = 1.272966$ and $\ln(1.009)\approx0.00896$. Then $1.272966 = 4t\times0.00896$. $4t=\frac{1.272966}{0.00896}\approx142.184$. $t=\frac{142.184}{4}\approx35.546\approx36$ years.

Answer:

D. 36 years