2026 NASS P1 Q7

The diagram shows an open cylinder tank with \(O\) as the centre of the base and \(0^\circ<\theta<90^\circ\). It is given that angle \(EOF=2\theta\) and \(OF=5\) cm.

1. Prove that the total surface area of the cylinder is \(S=\dfrac{25\pi}2(2\sin2\theta-\cos2\theta+1)\).[4]
2. Express \(S\) in the form \(\dfrac{25\pi}2\bigl(R\sin(2\theta-\alpha)+1\bigr)\), where \(R>0\) and \(\alpha\) is an acute angle.[3]
3. Hence find the maximum total surface area of the cylindrical tank and the corresponding value of \(\theta\).[2]