an investor puts $12,800 into an account that is compounded monthly. if the rate of return is 8%, the…

an investor puts $12,800 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 $50,000. use the formula a = p(1 + \\frac{r}{n})^{nt}

an investor puts $12,800 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 $50,000. 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 $A = 50000$, $P = 12800$, $r=0.08$ (8% annual interest rate), and $n = 12$ (compounded monthly).

Step2: Substitute the values into the formula

$50000=12800(1 +\frac{0.08}{12})^{12t}$. First, divide both sides by 12800: $\frac{50000}{12800}=(1+\frac{0.08}{12})^{12t}$. $\frac{50000}{12800}=\frac{125}{32}\approx3.90625$. And $1+\frac{0.08}{12}=1+\frac{1}{150}=\frac{151}{150}\approx1.00667$. So, $3.90625 = (\frac{151}{150})^{12t}$.

Step3: Take the natural logarithm of both sides

$\ln(3.90625)=12t\ln(\frac{151}{150})$. We know that $\ln(3.90625)\approx1.362$, and $\ln(\frac{151}{150})\approx\ln(1.00667)\approx0.00665$.

Step4: Solve for $t$

$t=\frac{\ln(3.90625)}{12\ln(\frac{151}{150})}$. $12\ln(\frac{151}{150})\approx12\times0.00665 = 0.0798$. $t=\frac{1.362}{0.0798}\approx17$.

Answer:

17