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Mathematics

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.

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Answer

(i) Given,

Area of base floor = 154 m2

πr2=154227×r2=154r2=154×722r2=7×7r=7 m.\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

13πr2h=123213×227×(7)2×h=12321543×h=1232h=1232×3154h=24 m.\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 = 625\sqrt{625}

⇒ l = 25 m.

Curved surface area of cone = πrl

= 227×7×25\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 = 5502\dfrac{550}{2}

⇒ l = 275 m.

Hence, length of canvas required = 275 m.

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