10 A square sheet of metal has edges 30 cm long.
Four squares each with edge \(x \mathrm {~cm}\), where \(x < 15\), are removed from the corners of the sheet.
The four rectangular sections are bent upwards to form an open-topped box, as shown in the diagrams.
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_392_460_630_347}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_387_437_635_872}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_282_380_703_1318}
10
- Show that the capacity, \(C \mathrm {~cm} ^ { 3 }\), of the box is given by
$$C = 900 x - 120 x ^ { 2 } + 4 x ^ { 3 }$$
10
- Find the maximum capacity of the box.
Fully justify your answer.