Question

A conical tent has the area of the base as 154m2 and its curved surface area as 550m2 , then the volume of the tent is :

Answer 1

Given :-

• A conical tent has the area of the base as 154m² and its curved surface area as 550m² .

To Find :

• Volume of tent = ?

Solution :

Let radius of conical tent be 'r' , height be 'h' and slant height be 'l'.

★ Finding radius of conical tent :

→ Base area of conical tent = πr²

→ 154 = 22/7 × r²

→ 154 × 7 = 22 × r²

→ 1078 = 22 × r²

→ 1078 ÷ 22 = r²

→ 49 = r²

→ r² = 49

→ r = √49

→ r = 7 m

⛬ Radius of the conical tent is 7 m.

★ Finding Slant Height of conical tent :

➻ C.S.A of cone = πrl

➻ 550 = 22/7 × 7 × l

➻ 550 = 22 × l

➻ 550 ÷ 22 = l

➻ 25 = l

➻ l = 25 m

⛬ Slant Height of the conical tent is 25 m.

★ Finding Height of conical tent :

➺ (Slant Height)² = (Radius)² + (Height)²

➺ (l)² = (r)² + (h)²

➺ (25)² = (7)² + (h)²

➺ 625 = 49 + h²

➺ 625 - 49 = h²

➺ 576 = h²

➺ h = 576

➺ h = √576

➺ h = 24 m

⛬ Height of the conical tent is 24 m.

★ Finding Volume of conical tent :

⊷ Volume of cone = ⅓ πr²h

⊷ Volume of cone = ⅓ × 22/7 × (7)² × 24

⊷ Volume of cone = ⅓ × 22/7 × 49 × 24

⊷ Volume of cone = ⅓ × 22 × 7 × 24

⊷ Volume of cone = 22 × 7 × 8

⊷ Volume of cone = 1232 m³

⛬ Volume of the conical tent is 1232 m³.

Answer 2

Given :

• Area of the base of conical tent = 154 m²
• Curved surface area = 550 m²

To Find :

• Volume of the tent

Solution :

• Let us find the radius of the base of tent :

• Area of base = πr²

$\leadsto \sf Base\:of\:radius\:=\:π{r}^{2}$

$\leadsto \sf 154\:=\: \dfrac{22}{7} \times {r}^{2}$

$\leadsto \sf {r}^{2}\:=\: \dfrac{154 \times 7}{22}$

$\leadsto \sf {r}^{2}\:=\: \dfrac{1078}{22}$

$\leadsto \sf {r}^{2}\:=\:49$

$\leadsto \sf r\:=\: \sqrt{49}$

$\bigstar \: \large{\sf{\underline{\red{Radius\:=\: 7\:m}}}}$

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• Now , let us find slant height ' l ' of the conical tent :

• CSA of cone = πrl

$\leadsto \sf 550\:=\: \dfrac{22}{7} \times 7 \times l$

$\leadsto \sf l\:=\: \dfrac{550 \times 7}{ 22 \times 7}$

$\leadsto \sf l\:=\: \dfrac{550}{22}$

$\bigstar \: \large{\bf{\underline{\red{Slant\:height \:=\: 25\:m}}}}$

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• Now , let us find height of conical tent :

$\leadsto \sf {(Slant\:height)}^{2}\:=\:{(radius)}^{2}\:+\:{(height)}^{2}$

$\leadsto \sf {(25)}^{2}\:=\: {(7)}^{2}\:+\:{(height)}^{2}$

$\leadsto \sf {(height)}^{2}\:=\:625\:-\:49$

$\leadsto \sf {(height)}^{2}\:=\: 576$

$\leadsto \sf height\:=\: \sqrt{576}$

$\bigstar \: \large{\sf{\underline{\red{Height\:=\:24\:m}}}}$

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• Now , let us find the volume of conical tent :

$\bullet \: \large {\boxed{\sf{\red{Volume\:of\:cone\:=\: \dfrac{1}{3} \:π{r}^{2} h}}}}$

$\leadsto \sf Volume\:=\: \dfrac{1}{3} \times \dfrac{22}{7} \times {7}^{2} \times 24$

$\leadsto \sf Volume\:=\: \dfrac{1}{3} \times \dfrac{22}{7} \times 49 \times 24$

$\leadsto \sf Volume\:=\:22 \times 7 \times 8$

$\leadsto \sf Volume\:=\: 22 \times 56$

$\bigstar \: \large{\boxed{\sf{\pink{Volume\:=\:1232\:{m}^{3}}}}}$

• Volume of the conical tent is 1232 m³

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Answer 3

Given :

• Area of the base of conical tent = 154 m²
• Curved surface area = 550 m²

To Find :

• Volume of the tent

Solution :

• Let us find the radius of the base of tent :

• Area of base = πr²

$\leadsto \sf Base\:of\:radius\:=\:π{r}^{2}$

$\leadsto \sf 154\:=\: \dfrac{22}{7} \times {r}^{2}$

$\leadsto \sf {r}^{2}\:=\: \dfrac{154 \times 7}{22}$

$\leadsto \sf {r}^{2}\:=\: \dfrac{1078}{22}$

$\leadsto \sf {r}^{2}\:=\:49$

$\leadsto \sf r\:=\: \sqrt{49}$

$\bigstar \: \large{\sf{\underline{\red{Radius\:=\: 7\:m}}}}$

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• Now , let us find slant height ' l ' of the conical tent :

• CSA of cone = πrl

$\leadsto \sf 550\:=\: \dfrac{22}{7} \times 7 \times l$

$\leadsto \sf l\:=\: \dfrac{550 \times 7}{ 22 \times 7}$

$\leadsto \sf l\:=\: \dfrac{550}{22}$

$\bigstar \: \large{\bf{\underline{\red{Slant\:height \:=\: 25\:m}}}}$

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• Now , let us find height of conical tent :

$\leadsto \sf {(Slant\:height)}^{2}\:=\:{(radius)}^{2}\:+\:{(height)}^{2}$

$\leadsto \sf {(25)^{2}\:=\: {(7)}^{2}\:+\:{(height)}^{2}$

$\leadsto \sf {(height)}^{2}\:=\:625\:-\:49$

$\leadsto \sf {(height)}^{2}\:=\: 576$

$\leadsto \sf height\:=\: \sqrt{576}$

$\bigstar \: \large{\sf{\underline{\red{Height\:=\:24\:m}}}}$

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• Now , let us find the volume of conical tent :

$\bullet \: \large {\boxed{\sf{\red{Volume\:of\:cone\:=\: \dfrac{1}{3} \:π{r}^{2} h}}}}$

$\leadsto \sf Volume\:=\: \dfrac{1}{3} \times \dfrac{22}{7} \times {7}^{2} \times 24$

$\leadsto \sf Volume\:=\: \dfrac{1}{3} \times \dfrac{22}{7} \times 49 \times 24$

$\leadsto \sf Volume\:=\:22 \times 7 \times 8$

$\leadsto \sf Volume\:=\: 22 \times 56$

$\bigstar \: \large{\boxed{\sf{\pink{Volume\:=\:1232\:{m}^{3}}}}}$

• Volume of the conical tent is 1232 m³

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