2025 TJC Promo Q5

[It is given that the arc length and the area of a sector with radius \(r\) and angle \(\theta\) (measured in radians) are given by \(r\theta\) and \(\dfrac{1}{2}r^2\theta\) respectively.]

The figure below shows a closed box designed by a pizza shop to pack slices of pizza. The box is a right prism of height \(h\) cm and its cross-section is a sector of a circle with radius \(r\) cm and an angle of 0.8 radians.

It is given that both \(r\) and \(h\) are allowed to vary. The box is to have a fixed volume of 200 cm\(^3\).

1. Show that the total surface area of the box, \(S\) cm\(^2\), is given by[4]
   \(S=\dfrac{4}{5}r^2+\dfrac{1400}{r}.\)
2. Use differentiation to find the value of \(r\) for which \(S\) is minimum.[3]
3. Determine whether it is possible for the shop to use such boxes with the minimum surface area to pack slices from a 12-inch (\(\approx 30.48\) cm in diameter) pizza.[1]