raina needs $35,112 for a future project. she can invest $6000 now at an annual rate of 11%, compounded…

Explanation:

Step1: Identify 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 compounded per year, and $t$ is the number of years. Here, $A = 35112$, $P = 6000$, $r=0.11$, and $n = 2$ (compounded semiannually).

Step2: Substitute values into the formula

Substitute the values into the formula: $35112=6000(1 +\frac{0.11}{2})^{2t}$. First, simplify the inside of the parentheses: $1+\frac{0.11}{2}=1 + 0.055=1.055$. So the equation becomes $\frac{35112}{6000}=(1.055)^{2t}$.

Step3: Simplify the left - hand side

$\frac{35112}{6000}=5.852$, so $5.852=(1.055)^{2t}$.

Step4: Take the natural logarithm of both sides

$\ln(5.852)=\ln((1.055)^{2t})$. Using the property of logarithms $\ln(a^{b})=b\ln(a)$, we get $\ln(5.852)=2t\ln(1.055)$.

Step5: Solve for $t$

We know that $\ln(5.852)\approx1.766$ and $\ln(1.055)\approx0.0536$. So, $1.766 = 2t\times0.0536$. First, simplify the right - hand side: $2t\times0.0536 = 0.1072t$. Then, $t=\frac{1.766}{0.1072}$. $t\approx16.47$.

Answer:

$16.47$