Question

(vii) A conical tent is required to accommodate 5 people each person must have 16 sq-ft of space on
the ground and 100 cubic-ft of air to breathe. Find out the vertical height and slant height of the tent.​

Answer 1

${\huge{\rm{\underline{\underline{Answer\checkmark}}}}}$

$\frak{Given}\begin{cases}\rm{\bf{5\:pearsons}\:are \:to\:be\:accomodated.}\\\rm{Space\:on\:the\:ground\:required\:by\:each\:person=\bf{16\:ft^2}}\\ \rm{Air\:required\:by\:each\:person=\bf{100\:ft^3}}\end{cases}$

${\underline{\underline{\frak{\dag\:To\:Find:-}}}}$

• The height of the tent.
• The slant height of the tent.

${\underline{\underline{\frak{\dag\:Required\:Solution:-}}}}$

Refer to the attachment 1 and 2 ...

So ,

$\colon \implies \sf \: volume _{ \: ( \: tent \: )} = 5 \times air \: requied \: by \: each \: person \\ \\ \colon \implies \: \sf \frac{1}{3} \pi \: {r}^{2} h = 5 \times 100 \: {ft}^{2} \\ \\ \sf \: we \: know \: that \: ( \pi {r}^{2} = 80 \: {ft}^{2} ) \: so \\ \\ \colon \implies \sf \: \frac{1}{3} \times \cancel{80} \: \cancel{ft}^{2} \times h = 5 \times \cancel{100} \: \cancel {ft}^{3} \\ \\ \colon \implies \sf \: 4 \times h = 5 \times 5 \times 3 \: ft \\ \\ \longrightarrow \underline{ \boxed{ \tt \purple{height \: ( \: h \: ) = \bf \frac{75}{4} \: ft}}}.... \sf(ii)$

Now , we know,

$\mapsto \rm \: slant \: height {}^{2} = radius {}^{2} + height {}^{2}$

Here ,

$\colon \implies \sf \: {l}^{2} = {r}^{2} + {h}^{2} \\ \\ \sf \: from \: eq(i) \: and \: eq(ii) \: \\ \\ \colon \implies \: \sf {l}^{2} = \frac{280}{11} + {( \frac{75}{4}) }^{2} \\ \\ \colon \implies \: \sf {l}^{2} = \frac{280}{11} + \frac{5625}{16} \\ \\ \colon \implies \sf\: l = \sqrt{ \frac{4480 + 61875}{176} } \\ \\ \colon \implies \sf \: l = \sqrt{ \frac{66355}{176} } \\ \\ \colon \implies \sf \: l = \sqrt{377.01} \\ \\ \longrightarrow \: \underline{ \boxed{ \tt \pink{slant \: height \: (l) \: = \bf 19.4 \: ft}}}$

So ,

Final Answer :

• Straight height = 75/4 ft = 18.75 ft
• Slant height = 19.4 ft.

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★ More About Cone :

• It is a 3 dimensional shape.
• It has a circular base.
• Some formulas related to cone are :

$\boxed{ \begin{array}{ c } \underline{\bf \star\: formulas _{ \: cone \: } \: : } \\ \\ \bull \sf \: volume \: = \: \dfrac{1}{3} \pi {r}^{2} h \\ \\ \bull \sf \: csa \: = \: \pi \: rl \: \\ \\ \bull \sf \: tsa \: = \: \pi {r}^{2} + \pi \: rl \\ \\ \sf \bull \: {l}^{2} = {r}^{2} + {h}^{2} \end{array}}$

Here,

• r = radius of base
• h = height
• l = slant height.

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