Question

1.] The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of 210 per 100 m² .
2.] A joker's cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.
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Answer 1

$\qquad \: \: \: \: \large{\pmb{\underline{\underline{\frak{ Question \: 1 }}}}}$

Given :

• Slant Height of the tomb = 25 m
• Base Diameter of tomb = 14 m
• Rate of white - washing = Rs.210 per 100 m²

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To Find :

• Cost of white - washing = ?

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Solution :

~ Formula Used :

${\orange{\dashrightarrow}} \: \: {\underline{\boxed{\purple{\sf{ Curved \: Surface \: Area{\small_{(Cone)}} = \pi rl }}}}}$

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~ Calculating the Curved Surface Area :

${\longmapsto{\qquad{\sf{ CSA{\small_{(Conical \: Tomb)}} = \pi rl }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ CSA{\small_{(Conical \: Tomb)}} = \dfrac{22}{7} \times \dfrac{Diameter}{2} \times 25 }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ CSA{\small_{(Conical \: Tomb)}} = \dfrac{22}{7} \times \dfrac{14}{2} \times 25 }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ CSA{\small_{(Conical \: Tomb)}} = \dfrac{22}{7} \times \dfrac{\cancel{14}}{\cancel2} \times 25 }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ CSA{\small_{(Conical \: Tomb)}} = \dfrac{22}{\cancel7} \times \cancel7 \times 25 }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ CSA{\small_{(Conical \: Tomb)}} = 22 \times 25 }}}} \\ \\ \ {\qquad{\textsf{ Curved Surface Area of the tomb = {\green{\sf{ 550 \: m² }}}}}}$

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~ Calculating the Cost :

${\longrightarrow{\qquad{\sf{ Cost{\small_{(White - washing )}} = CSA \times Rate }}}} \\ \\ \ {\longrightarrow{\qquad{\sf{ Cost{\small_{(White - washing )}} = 550 \times \dfrac{210}{100} }}}} \\ \\ \ {\longrightarrow{\qquad{\sf{ Cost{\small_{(White - washing )}} = \dfrac{115500}{100} }}}} \\ \\ \ {\longrightarrow{\qquad{\sf{ Cost{\small_{(White - washing )}} = \cancel\dfrac{115500}{100} }}}} \\ \\ \ {\qquad{\textsf{ Cost of white - washing the Tomb = {\pink{\sf{ ₹ \: 1155 }}}}}}$

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Therefore :

❝ Cost of white - washing the tomb is ₹ 1155 . ❞

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$\qquad \: \: \: \: \large{\pmb{\underline{\underline{\frak{ Question \: 2 }}}}}$

Given :

• Radius of Cone = 7 cm
• Height of cone = 24 cm
• No. of Caps = 10

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To Find :

• Sheet Required for making 10 caps = ?

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Solution :

~ Formula Used :

• Pythagoras Theorem :

${\orange{\dashrightarrow}} \: \: {\underline{\boxed{\purple{\sf{ Slant \: Height² = Height² + Radius² }}}}}$

Denoted :

• Slant Height = l
• Height = h
• Radius = r

• Curved Surface Area :

${\orange{\dashrightarrow}} \: \: {\underline{\boxed{\purple{\sf{ Curved \: Surface \: Area{\small_{(Cone)}} = \pi rl }}}}}$

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~ Calculating the Slant Height :

${:\implies{\qquad{\sf{ l² = h² + r² }}}} \\ \\ \ {:\implies{\qquad{\sf{ l² = 24² + 7² }}}} \\ \\ \ {:\implies{\qquad{\sf{ l² = 576 + 49 }}}} \\ \\ \ {:\implies{\qquad{\sf{ l² = 625 }}}} \\ \\ \ {:\implies{\qquad{\sf{ l = \sqrt{625} }}}} \\ \\ \ {\qquad{\textsf{ Slant Height of the cap = {\blue{\sf{ 25 \: cm}}}}}}$

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~ Calcuting the Curved Surface Area :

${\longmapsto{\qquad{\sf{ CSA{\small_{(Cap)}} = \pi rl }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ CSA{\small_{(Cap)}} = \dfrac{22}{7} \times 7 \times 25 }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ CSA{\small_{(Cap)}} = \dfrac{22}{7} \times \cancel7 \times 25 }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ CSA{\small_{(Cap)}} = 22 \times 25 }}}} \\ \\ \ {\qquad{\textsf{ Curved Surface Area of the Cap = {\color{maroon}{\sf{ 550 \: cm² }}}}}}$

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~ Calcuting the Sheet Required :

${\longrightarrow{\qquad{\sf{ CSA{\small_{(10 \: Caps )}} = CSA{\small_{(1 \: Cap)}} \times No. \: of \: Caps }}}} \\ \\ \ {\longrightarrow{\qquad{\sf{ CSA{\small_{(10 \: Caps)}} = 550 \times 10 }}}} \\ \\ \ {\qquad{\textsf{ Area of Sheet Required = {\red{\sf{ 5500 \: cm² }}}}}}$

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Therefore :

❝ Sheet Required for making 10 Caps is 5500 cm² .❞

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