9.
\begin{figure}[h]
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\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{1425d933-47e3-4a12-bcab-fd2ca41827e2-4_799_1299_303_285}
\end{figure}
\includegraphics[max width=\textwidth, alt={}]{1425d933-47e3-4a12-bcab-fd2ca41827e2-4_303_1127_1144_338}
A rectangular sheet of metal measures 50 cm by 40 cm . Squares of side \(x \mathrm {~cm}\) are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 3.
- Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the tray is given by \(V = 4 x \left( x ^ { 2 } - 45 x + 500 \right)\).
- State the range of possible values of \(x\).
- Find the value of \(x\) for which \(V\) is a maximum.
- Hence find the maximum value of \(V\).
- Justify that the value of \(V\) you found in part (d) is a maximum.