Question

The curved surface area of a right circular cone is 12320 cm2. If the radius
its base is 56 cm, find its height.​

Answer 1

Given: The curved surface area of a right circular cone is 12320 cm², and the radius it's base is 56 cm.

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Need to find: The height of a right circular cone.

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We know that, if we are given with the curved surface area of a right circular cone and radius it's base, we have the required formula, that is,

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$\sf{:\implies CSA_{(cone)} = \pi rl}$

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⠀⠀⠀⠀ Here CSA is the curved surface area, r is the radius and l is the slant height, And here in this question we have CSA = 12320 cm² and r = 56 cm. So by using the required formula we can calculate the slant height of cone.

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By using the formula and substituting all the values in the formula, we get:

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$\sf{:\implies 12320 = \dfrac{22}{ \cancel{ \: 7 \: }} \times \cancel{56} \times l} \\ \\ \\ \sf{:\implies 12320 = 22 \times 8 \times l} \\ \\ \\ \sf{:\implies 12320 = 176 \times l} \\ \\ \\ \sf{:\implies 12320 = 176l} \\ \\ \\ \sf{:\implies l = \cancel{\frac{12320}{176} }} \\ \\ \\ \sf{:\implies \boxed{\frak{ \red{l = 70}}}}$

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Now,

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We know that, if we are given with the slant height & radius of the right circular cone, we have the required formula, that is,

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$\sf{:\implies {(l)}^{2} = {(r)}^{2} + {(h)}^{2} }$

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⠀⠀⠀⠀ Here l is the slant height, r is the radius, and h is the height of right circular cone, And here we have l = 70, r = 56. So by using the required formula we can find the height of right circular cone.

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By using the formula and substituting all the values in the formula, we get:

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$\sf{:\implies {(70)}^{2} = {(56)}^{2} + {(h)}^{2}} \\ \\ \\ \sf{:\implies 4900 = 3136 + {h}^{2}} \\ \\ \\ \sf{: \implies {h}^{2} = 4900 - 3136} \\ \\ \\ \sf{: \implies {h}^{2} = 1764} \\ \\ \\ \sf{: \implies h = \sqrt{1764} } \\ \\ \\ \sf{: \implies \boxed{ \frak{ \pink{h = 42}}}}$

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$\sf{\therefore\underline{The \; height \; of \; the \; cone \; is \; \textsf{\textbf{42 cm.}}}}$

Answer 2

Answer:

Given :-

Radius of the cone ( r ) = 56 cm

Let the height = h cm

Slant height = l cm

CSA of the cone = 12320 cm²

☯️ π rl = 12320

=> ( 22/7 ) × 56 × l = 12320

=> l = ( 12320 × 7 ) / ( 56 × 22 )

=> l = 70 cm

=> l² = r² + h²

=> h² = l² - r²

=> 70² - 56²

=> ( 70 + 56 ) ( 70 - 56 )

=> 126 × 14

=> 14 × 3 × 3 × 14

=> h = √ ( 14 × 14 ) × ( 3 × 3 )

=> h = 14 × 3

=> h = 42 cm