an investor puts $12,000 into an account that is compounded monthly. if the rate of return is 8%, the…

an investor puts $12,000 into an account that is compounded monthly. if the rate of return is 8%, the investor will need to wait approximately years until the amount in the account reaches $20,000. use the formula a = p(1 + \\frac{r}{n})^{nt}

an investor puts $12,000 into an account that is compounded monthly. if the rate of return is 8%, the investor will need to wait approximately years until the amount in the account reaches $20,000. use the formula a = p(1 + \\frac{r}{n})^{nt}

Answer

Explanation:

Step1: Identify the compound - interest formula variables

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A = 70000$, $P = 12000$, $r=0.08$ (8% annual interest rate), and $n = 12$ (compounded monthly).

Step2: Substitute the values into the formula

$70000=12000(1 +\frac{0.08}{12})^{12t}$. First, divide both sides by 12000: $\frac{70000}{12000}=(1+\frac{0.08}{12})^{12t}$, so $\frac{35}{6}=(1+\frac{0.08}{12})^{12t}$.

Step3: Simplify the right - hand side base

$1+\frac{0.08}{12}=1+\frac{1}{150}=\frac{151}{150}$. The equation becomes $\frac{35}{6}=(\frac{151}{150})^{12t}$.

Step4: Take the natural logarithm of both sides

$\ln(\frac{35}{6}) = 12t\ln(\frac{151}{150})$.

Step5: Solve for $t$

We know that $\ln(\frac{35}{6})\approx1.79176$ and $\ln(\frac{151}{150})\approx0.00665$. Then $t=\frac{\ln(\frac{35}{6})}{12\ln(\frac{151}{150})}$. $12\ln(\frac{151}{150})\approx12\times0.00665 = 0.0798$. $t=\frac{1.79176}{0.0798}\approx22.45$.

Answer:

22.45