7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-18_675_1408_292_358}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a rectangular sheet of metal of negligible thickness, which measures 25 cm by 15 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 cuboid box, as shown in Figure 2.
- Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the box is given by
$$V = 4 x ^ { 3 } - 80 x ^ { 2 } + 375 x$$
- Use calculus to find the value of \(x\), to 3 significant figures, for which the volume of the box is a maximum.
- Justify that this value of \(x\) gives a maximum value for \(V\).
- Find, to 3 significant figures, the maximum volume of the box.
\section*{8.}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-22_670_1004_292_392}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The curve crosses the \(y\)-axis at the point \(( 0,5 )\) and crosses the \(x\)-axis at the point \(( 6,0 )\).
The curve has a minimum point at \(( 1,3 )\) and a maximum point at \(( 4,7 )\).
On separate diagrams, sketch the curve with equation