A stage canopy support post at a music festival is capped with a cylinder of radius \(7\) cm and height \(H\) cm, topped with a cone of the same radius and ...

The support post's cap is a cylinder topped with a cone of the same radius; Pythagoras' theorem finds the cone's slant height from its radius and height, and the total volume equation (cylinder plus cone) can then be solved for the cylinder's unknown height \(H\).

(a) The cone's radius (7 cm), height (24 cm) and slant height form a right-angled triangle, so \(\text{slant}^{2}=7^{2}+24^{2}=49+576=625\). Taking the square root: slant \(=\sqrt{625}=25\) cm. [2 marks]

(b) The cylinder's volume is \(\pi(7)^{2}H=49\pi H\); the cone's volume is \(\dfrac13\pi(7)^{2}(24)=\dfrac13\pi(49)(24)=392\pi\). Adding and setting equal to the total: \(49\pi H+392\pi=882\pi\). Dividing every term by \(\pi\): \(49H+392=882\), so \(49H=490\) and \(H=10\) cm. [3 marks]

(c) The curved surface area is the cylinder's curved surface plus the cone's curved surface: \(2\pi(7)(10)+\pi(7)(25)=140\pi+175\pi=315\pi\) cm\(^{2}\). [1 mark]

(d) Evaluating \(315\pi\approx989.6\) cm\(^{2}\), which is less than \(1000\) cm\(^{2}\), so one roll of canopy material is enough to cover one post cap. [1 mark]

The cone's curved surface area formula, \(\pi r l\), uses the slant height \(l\) found in part (a), not the vertical height of 24 cm; mixing these two lengths up is a common error when a cone's surface area follows a "show that" height question.