2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12fbfe89-60fe-4890-9a22-2b1988d05d33-03_424_465_228_721}
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\caption{Figure 1}
\end{figure}
Figure 1 shows a metal cube which is expanding uniformly as it is heated. At time \(t\) seconds, the length of each edge of the cube is \(x \mathrm {~cm}\), and the volume of the cube is \(V \mathrm {~cm} ^ { 3 }\).
- Show that \(\frac { \mathrm { d } V } { \mathrm {~d} x } = 3 x ^ { 2 }\)
Given that the volume, \(V \mathrm {~cm} ^ { 3 }\), increases at a constant rate of \(0.048 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\),
- find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\), when \(x = 8\)
- find the rate of increase of the total surface area of the cube, in \(\mathrm { cm } ^ { 2 } \mathrm {~s} ^ { - 1 }\), when \(x = 8\)