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The dimensions of a cuboid are \(x \mathrm {~cm} , 2 x \mathrm {~cm}\) and \(4 x \mathrm {~cm}\), as shown in the diagram.
- Show that the surface area \(S \mathrm {~cm} ^ { 2 }\) and the volume \(V \mathrm {~cm} ^ { 3 }\) are connected by the relation
$$S = 7 V ^ { \frac { 2 } { 3 } }$$
- When the volume of the cuboid is \(1000 \mathrm {~cm} ^ { 3 }\) the surface area is increasing at \(2 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of the volume at this instant.