An initial $1800 investment was worth $2299.16 after two years and nine months. What semi-annually compounded nominal rate of return did the investment earn?

To find the semi-annually compounded nominal rate of return, we can use the formula for compound interest:

A = P \left(1 + \frac{r}{n}\right)^{nt}

where:
- A = \ is the future value after 2\frac{3}{4} years
- P = \ is the initial investment
- r is the nominal annual interest rate we want to find
- n = 2 is the number of compounding periods per year (semi-annually)
- t = 2\frac{3}{4} years

Substitute the given values into the formula to solve for r :

\1800 \left(1 + \frac{r}{2}\right)^{2\frac{3}{4} \cdot 2}

Simplify the equation:

\frac{2299.16}{1800} = \left(1 + \frac{r}{2}\right)^\frac{11}{2}

\frac{1277.31}{1000}=\left(1+\frac{r}{2}\right)^{\frac{11}{2}}

Now, solve for r :

\left(1+\frac{r}{2}\right)=\sqrt[\frac{11}{2}]{\frac{1277.31}{1000}}

\frac{r}{2}=\sqrt[\frac{11}{2}]{\frac{1277.31}{1000}}-1

r=2\left(\sqrt[\frac{11}{2}]{\frac{1277.31}{1000}}-1\)]Calculating\right. r :\[r\approx2(\sqrt[\frac{11}{2}]{1.27731}-1)\]

r\approx2(\approx0.0455)

r\approx\boxed{0.091}

\textbf{Answer:} The semi-annually compounded nominal rate of return the investment earned is 9.1\% .