Answer
The dimensions of the piece of sheet metal produced by the machine will be $23\text{cm}\times 23\text{cm}$.
Work Step by Step
The machine produces open boxes by the use of square sheets of metal.
We use the volume of square sheets of metal to find out the side of the square boxes
$V={{x}^{3}}$
The machine cuts equal sizes of squares of side $9$ centimeters.
Therefore, the volume of the square sheets of metal is
$V=9{{x}^{2}}$
And the volume of each box is $225$ cubic centimeters,
$\begin{align}
& V=9{{x}^{2}}=225 \\
& {{x}^{2}}=\frac{225}{9} \\
& =25 \\
& x=\pm 5
\end{align}$
The length of the side can not be negative, so
$x=5$
Hence, the dimensions of the sheet metal are found as:
$\begin{align}
& x+2\left( 9 \right)=5+18 \\
& =23
\end{align}$
The machine cuts equal sized squares, so, both values are same.
Therefore, the dimensions of the piece of sheet metal produced by the machine will be $23\text{cm}\times 23\text{cm}$.
