Question

The curved surface area of a cone is 4070cm² and it's diameter is 70cm. find its height and total surface area.(takeπ=22/7)​

Answer 1

Answer :-

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

Here the concept of CSA of Cone and TSA of cone has been used. We are given the CSA and Diameter. From that we can find the Slant height. And from the we can find the Height. And after finding the height, we can apply values and find TSA.

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★ Formula Used :-

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

$\: \\ \large{\boxed{\sf{L^{2}\;=\;\bf{H^{2}\;+\;R^{2}}}}}$

$\: \\ \large{\boxed{\sf{TSA\;of\;Cone\;=\;\bf{CSA\;+\; \pi r^{2}}}}}$

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★ Question :-

The curved surface area of a cone is 4070cm² and it's diameter is 70cm. Find its height and total surface area.

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★ Solution :-

Given,

» CSA of Cone = 4070 cm²

» Diameter of Cone = d = 70 cm

» Radius of Cone = r = ½ × d = ½ × 70 = 35 cm

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~ For the Slant Height (L) of Cone :-

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

$\: \\ \qquad \large{\sf{:\Longrightarrow\;\;\; 4070\;cm^{2}\;=\;\bf{\dfrac{22}{7}\:\times\: 35\:\times\: L}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow\;\;\; L\;\;=\;\bf{\dfrac{\: 4070}{22\:\times\:5}\:\: = \: \: \underline{\underline{37\:cm}}}}}$

$\: \\ \large{\boxed{\boxed{\tt{Slant\;\;Height\;\;of\;\;Cone\;\;=\;\;\bf{35\;\;cm}}}}}$

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~ For the Height of the Cone (H) :-

$\; \\ \large{\sf{:\longrightarrow\;\;\; L^{2}\;=\;\bf{H^{2}\;+\;R^{2}}}}$

$\; \\ \large{\sf{:\longrightarrow\;\;\; (37)^{2}\;=\;\bf{H^{2}\;+\;(35)^{2}}}}$

$\; \\ \large{\sf{:\longrightarrow\;\;\; 1369\;=\;\bf{H^{2}\;+\;1225}}}$

$\; \\ \large{\sf{:\longrightarrow\;\;\; H^{2}\;=\;\bf{1369\;-\;1225}}}$

$\; \\ \large{\sf{:\longrightarrow\;\;\; H^{2}\;=\;\bf{\sqrt{144}\;\:=\;\;\underline{\underline{12\;\;cm}}}}}$

$\: \\ \large{\boxed{\boxed{\tt{Height\;\;of\;\;Cone\;\;=\;\;\bf{12\;\;cm}}}}}$

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~ For TSA of Cone :-

$\: \\ \qquad \large{\sf{:\Longrightarrow\;\;\; TSA\;of\;Cone\;=\;\bf{\pi rL\;+\; \pi r^{2}}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow\;\;\; TSA\;of\;Cone\;=\;\bf{CSA\;+\; \dfrac{22}{7}\;\times\;(35)^{2}}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow\;\;\; TSA\;of\;Cone\;=\;\bf{CSA\;+\; \dfrac{22}{7}\;\times\;1225}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow\;\;\; TSA\;of\;Cone\;=\;\bf{4070\;+\; (22\;\times\;175)}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow\;\;\; TSA\;of\;Cone\;=\;\bf{4070\;+\; 3850\;\;=\;\;\underline{\underline{7920\;\;cm^{2}}}}}}$

$\: \large{\boxed{\boxed{\tt{TSA\;\;of\;\;Cone\;\;=\;\;\bf{7920\;\;cm^{2}}}}}}$

$\: \\ \large{\underline{\underline{\rm{Thus,\;Height\;of\;cone\;is\;\boxed{\bf{12\;\;cm}}\;and\;TSA\;of\;Cone\;is\;\;\boxed{\bf{7920\;\;cm^{2}}}}}}}$

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$\: \large{\underline{\sf{More\;Formulas\;to\;know\;:-}}}$

• Volume of Cone = ⅓ × πr²h

• Volume of Cube = (Side)³

• Volume of Cuboid = Length × Breadth × Height

• Volume of Cylinder = πr²h

• Volume of Hemisphere = ⅔ × πr³

Answer 2

Given,

Curved surface area of cone is 4070 cm² and diameter of cone is 70 cm. π = 22/7

To find :

Height and total surface area of cone.

Solution :

Diameter of cone = 70 cm

⇒ Radius of cone = 70/2 cm

⇒ Radius of cone = 35 cm

Now we know,

⇒ CSA of cone = πrl

⇒ 4070 = 22/7 * 35 * l

⇒ 4070 = 22 * 5 * l

⇒ 4070 = 110 * l

⇒ l = 4070/110

⇒ l = 37 cm

Now we know,

⇒ Slant height, l = √(h² + r²)

⇒ 37 = √(h² + 35²)

⇒ 37 = √(h² + 1225)

⇒ 37² = h² + 1225

⇒ 1369 = h² + 1225

⇒ 1369 - 1225 = h²

⇒ h² = 144

⇒ h = √144

⇒ h = 12 cm

∴ Height of cone = 12 cm

Now we know,

⇒ TSA of cone = πr(l + r)

⇒ TSA of cone = 22/7 * 35(37 + 35)

⇒ TSA of cone = 22 * 5(72)

⇒ TSA of cone = 110 * 72

⇒ TSA of cone = 7920 cm²

Therefore,

Height of cone = 12 cm.

Total surface area of cone = 7920 cm².