Answer
12cm by 15cm.
Work Step by Step
Step 1. Assume the sides of the rectangle are $x$ cm and $y$ cm, with the given conditions, we can set up the following system of equations:
\begin{cases} xy=180 \\ 2(x+y)=54 \end{cases}
Step 2. From the first equation, we get $y=\frac{180}{x}$, substitute it in the second equation to get:
$2(x+\frac{180}{x})=54$ or $x^2-27x+180=0$
Step 3. Solve the above quadratic equation with factoring or a formula:
$(x-12)(x-15)=0$ and we get $x=12cm or 15cm$
Step 4. Back-substitute the results into the first equation, we get $x=12cm, y=15cm$ or $x=15cm, y=12cm$
Step 5. We conclude that the sizes of the rectangle are 12cm by 15cm.
