9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9215e382-406c-41a3-8907-f465b134dd87-4_499_1137_954_319}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a tray made from sheet metal.
The horizontal base is a rectangle measuring \(8 x \mathrm {~cm}\) by \(y \mathrm {~cm}\) and the two vertical sides are trapezia of height \(x \mathrm {~cm}\) with parallel edges of length \(8 x \mathrm {~cm}\) and \(10 x \mathrm {~cm}\). The remaining two sides are rectangles inclined at \(45 ^ { \circ }\) to the horizontal.
Given that the capacity of the tray is \(900 \mathrm {~cm} ^ { 3 }\),
- find an expression for \(y\) in terms of \(x\),
- show that the area of metal used to make the tray, \(A \mathrm {~cm} ^ { 2 }\), is given by
$$A = 18 x ^ { 2 } + \frac { 200 ( 4 + \sqrt { 2 } ) } { x } ,$$
- find to 3 significant figures, the value of \(x\) for which \(A\) is stationary,
- find the minimum value of \(A\) and show that it is a minimum.