Question

A conical tent is to accommodate 77persons. Each person must have 16m3

of air to breathe. Given

the radius of the tent is 7m, find the height of the tent and also its curved surface area.​

Answer 1

Answer :-

$\: \\ \large{\underline{\underline{\sf{\leadsto \: \: \: Firstly,\; let's \; understand \; the \; concept \; :-}}}}$

Here the concept of Volume of Cone has been used. We are given that the conical tent can accommodate 77 persons and each person needs 16 m³ air. So their product can give the volume of tent. From that we can find the height if cone and apply in the formula of CSA of Cone.

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★ Formula Used :-

$\: \\ \large{\boxed{\boxed{\sf{Volume \: of \: Cone \: = \: \bf{\dfrac{1}{3} \: \times \: \pi r^{2}h}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{Volume \: of \: Cone \: = \: \bf{Number \: of \: People \: Accomodated \; \times \; Volume \: of \: Air \: required \: by \: each}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{(Slant \: Height)^{2} , \; L^{2} \: \: = \: \: \bf{r^{2} \; + \; h^{2}}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{CSA \: of \: Cone \: = \: \: \bf{\pi rL}}}}}$

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★ Question :-

A conical tent is to accommodate 77 persons. Each person must have 16m³ of air to breathe. Given the radius of the tent is 7m, find the height of the tent and also its curved surface area.

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★ Solution :-

Given,

» Number of people tent can accommodate = 77

» Air required to breathe for each person = 16 m³

» Radius of the tent = r = 7 m

• Let the height of the conical tent be 'h' m.

Then, according to the question :-

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~ For the Height of the Conical Tent :-

$\: \\ \qquad \large{\sf{:\longrightarrow \: \: \: Volume \: of \: Cone \: = \: \bf{\dfrac{1}{3} \: \times \: \pi r^{2}h}}}$

$\: \\ \large{\sf{:\longrightarrow \: Number \: of \: People \: Accomodated \; \times \; Volume \: of \: Air \: required \: by \: each \: = \: \bf{\dfrac{1}{3} \: \times \: \pi r^{2}h}}}$

$\: \\ \qquad \large{\sf{:\longrightarrow \: \: \: 77 \: \times \: 16 \: m^{3} \: = \: \bf{\dfrac{1}{3} \: \times \: \dfrac{22}{\cancel{7}} \: \times \: (7\; m)^{\cancel{2}}\: \times \: h}}}$

$\: \\ \qquad \large{\sf{:\longrightarrow \: \: \: 1232 \: m^{3} \: = \: \bf{\dfrac{1}{3} \: \times \: 22 \: \times \: 7\:m^{2}\: \times \: h}}}$

$\: \\ \qquad \large{\sf{:\longrightarrow \: \: \: h \: = \: \bf{\dfrac{\: 1232 \: m^{\cancel{3}} \; \times \; 3}{\: 22\; \times \; 7 \: \cancel{m^{2}}} \: \: = \: \: 8\; m \: \times \: 3 \;\; = \: \: \underline{\underline{24 \: m}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{Height \; of \; the \; Conical \; Tent \; \: = \; \bf{24 \; m}}}}}$

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~ For the Slant Height of the Tent :-

$\: \\ \qquad \large{\sf{:\rightarrow \: \: \: Slant \: Height, \; L^{2} \: \: = \: \: \bf{r^{2} \; + \; h^{2}}}}$

$\: \\ \qquad \large{\sf{:\rightarrow \: \: \: Slant \: Height, \; L^{2}\: \: = \: \: \bf{(7\; m) ^{2} \; + \; (24\;m)^{2} \: \: = \: \: 49\; m^{2} \: + \: 576\; m^{2}}}}$

$\: \\ \qquad \large{\sf{:\rightarrow \: \: \: Slant \: Height, \; L^{2} \: \: = \: \: \bf{625 \; m^{2}}}}$

$\: \\ \qquad \large{\sf{:\rightarrow \: \: \: Slant \: Height, \; L \: \: = \: \: \bf{\sqrt{625 \; m^{2}} \: \: = \: \: \underline{\underline{25 \; m}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{Slant \; height \; of \; the \; Conical \; Tent \; \: = \; \bf{25 \; m}}}}}$

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~ For the CSA of Tent :-

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: CSA \: of \: Cone \: = \: \: \bf{\pi rL}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: CSA \: of \: Cone \: = \: \: \bf{\dfrac{22}{\cancel{7}} \: \times \: \cancel{7} \: m \: \times \: 25 \: m \: \: = \: \: \underline{\underline{550 \; m^{2}}}}}}$

$\: \\ \large{\underline{\underline{\rm{\mapsto \: \: \: Thus, \; the \; CSA \; of \; the \; Conical \; tent \; is \;\; \boxed{\bf{550 \;\; m^{2}}}}}}}$

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$\: \\ \large{\underbrace{\underbrace{\sf{More \; formulas \; to \; know \; \; :-}}}}$

• Volume of Cylinder = πr²h

• Volume of Cube = (Side)³

• Volume of Cuboid = Length(L) × Breadth(B) × Height(H)

• Volume of Sphere = ⅔ × πr³

• TSA of Cone = πr² + πrl

Answer 2

Given,

A conical tent is to accommodate 77persons. Each person must have 16m³ of air to breathe. Given  the radius of the tent is 7m.

To find :

Find the height of the tent and also its curved surface area.​

Solution :

Conical tent can accommodate 77 persons.

each person must have 16 m³ of air to breathe.

∴ Volume of conical tent = 77 * 16

∴ Volume of conical tent = 1232 m³

Now we know,

⇒ Volume of cone = 1/3 πr²h

⇒ 1232 = 1/3 * 22/7 * 7² * h

⇒ 1232 = 1/3 * 22 * 7 * h

⇒ 1232 = 51.33 * h

⇒ h = 1232/51.33

⇒ h = 24 cm

∴ Height of tent = 24 cm

Now we know,

⇒ Slant height(l) = √(h² + r²)

⇒ l = √(24² + 7²)

⇒ l = √625

⇒ l = 25 cm

Now we know,

⇒ CSA of cone = πrl

⇒ CSA of tent = 22/7 * 7 * 25

⇒ CSA of tent = 22 * 25

⇒ CSA of tent = 550 cm²

∴ Curved surface area of tent = 550 cm²