The volume of a conical tent is 1232 m³ and the area of the base floor is 154 m². Calculate the :

(i) radius of the floor,

(ii) height of the tent,

(iii) length of the canvas required to cover this conical tent if its width is 2 m.

(i) Given,

Area of base floor = 154 m²

$\Rightarrow πr^2 = 154 \\[1em] \Rightarrow \dfrac{22}{7} \times r^2 = 154 \\[1em] \Rightarrow r^2 = \dfrac{154 \times 7}{22} \\[1em] \Rightarrow r^2 = 7 \times 7 \\[1em] \Rightarrow r = 7 \text{ m}.$

Hence, radius of the floor = 7 m.

(ii) Given,

Volume of tent = 1232 m³

$\Rightarrow \dfrac{1}{3}πr^2h = 1232 \\[1em] \Rightarrow \dfrac{1}{3} \times \dfrac{22}{7} \times (7)^2 \times h = 1232 \\[1em] \Rightarrow \dfrac{154}{3} \times h = 1232 \\[1em] \Rightarrow h = \dfrac{1232 \times 3}{154} \\[1em] \Rightarrow h = 24 \text{ m}.$

Hence, height of tent = 24 m.

(iii) By formula,

⇒ l² = r² + h²

⇒ l² = (7)² + (24)²

⇒ l² = 49 + 576

⇒ l² = 625

⇒ l = $\sqrt{625}$

⇒ l = 25 m.

Curved surface area of cone = πrl

= $\dfrac{22}{7} \times 7 \times 25$

= 550 m².

Let length of canvas be l.

Area of canvas = Curved surface area of cone

⇒ l × b = 550

⇒ l × 2 = 550

⇒ l = $\dfrac{550}{2}$

⇒ l = 275 m.

Hence, length of canvas required = 275 m.