nathaniel invested $37,000 in an account paying an interest rate of 5.7% compounded daily. assuming no…

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 $P = 37000$, $A=70900$, $r = 0.057$ (since $5.7%=0.057$), and $n = 365$ (compounded daily). Substitute these values into the formula: $70900=37000(1 +\frac{0.057}{365})^{365t}$.

Step2: Simplify the equation

First, divide both sides of the equation by $37000$: $\frac{70900}{37000}=(1+\frac{0.057}{365})^{365t}$. $1.916216=(1 + 0.000156164)^{365t}$. $1.916216=(1.000156164)^{365t}$.

Step3: Take the natural logarithm of both sides

$\ln(1.916216)=\ln((1.000156164)^{365t})$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get: $\ln(1.916216)=365t\ln(1.000156164)$. We know that $\ln(1.916216)\approx0.640$ and $\ln(1.000156164)\approx0.00015615$.

Step4: Solve for $t$

$0.640 = 365t\times0.00015615$. $365t\times0.00015615=0.640$. $365t=\frac{0.640}{0.00015615}\approx4098.5$. $t=\frac{4098.5}{365}\approx11.23$.

Answer:

$11$ years