The volume of a right circular cone is 9856 cm3
.If the diameter of the base is 28cm ,find:
(i) height of the cone (ii) slant height of the cone (iii) curved surface area of the cone
Answers
Answer:
Height of the cone is 48 cm.
Slant height of the cone is 50 cm.
The curved surface area of the cone is 2200 cm².
Step-by-step-explanation:
We have given that,
Volume of cone is 9856 cm³.
Diameter of base of cone is 28 cm.
We have to find the height, slant height and curved surface are of the cone.
Now, we know that,
Volume of cone = ( π r² h ) / 3 - - [ Formula ]
⇒ 9856 = [ 22 / 7 * ( d / 2 )² * h ] / 3 - - [ ∵ r = d / 2 ]
⇒ 9856 = [ 22 / 7 * ( 28 / 2 )² * h ] / 3
⇒ 9856 = ( 22 / 7 * 28 / 2 * 28 / 2 * h ) / 3
⇒ 9856 = 22 / 7 * 28 / 2 * 28 / 2 * 1 / 3 * h
⇒ ( 9856 * 7 * 2 * 2 * 3 ) / ( 22 * 28 * 28 ) = h
⇒ 9856 ÷ 22 * 7 ÷ 28 * 2 ÷ 28 * 2 * 3 = h
⇒ h = 9856 ÷ 22 * 7 ÷ 28 * 2 ÷ 28 * 2 * 3
⇒ h = 448 * 1 ÷ 4 * 1 ÷ 14 * 2 * 3
⇒ h = 448 ÷ 4 * 1 ÷ 14 * 2 * 3
⇒ h = 112 * 1 ÷ 14 * 2 * 3
⇒ h = 112 ÷ 14 * 2 * 3
⇒ h = 8 * 2 * 3
⇒ h = 16 * 3
⇒ h = 48 cm
∴ Height of the cone is 48 cm.
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Now, we know that,
l² = h² + r² - - - [ Formula ]
⇒ l² = ( 48 )² + ( d / 2 )² - - [ ∵ r = d / 2 ]
⇒ l² = ( 24 * 2 )² + ( 28 / 2 )²
⇒ l² = ( 24 )² * ( 2 )² + ( 14 )²
⇒ l² = 576 * 4 + 196
⇒ l = √[ 576 * 4 + 196 ] - - [ Taking square roots ]
⇒ l = √( 2304 + 196 )
⇒ l = √2500
⇒ l = √( 50 * 50 )
⇒ l = 50 cm
∴ Slant height of the cone is 50 cm.
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Now, we know that,
Curved surface area of the cone = π r l - - [ Formula ]
⇒ CSAᶜᵒⁿᵉ = ( 22 / 7 ) * ( d / 2 ) * 50 - - [ ∵ r = d / 2 ]
⇒ CSAᶜᵒⁿᵉ = ( 22 / 7 ) * ( 28 / 2 ) * 50
⇒ CSAᶜᵒⁿᵉ = 22 ÷ 2 * 28 ÷ 7 * 50
⇒ CSAᶜᵒⁿᵉ = 11 * 4 * 50
⇒ CSAᶜᵒⁿᵉ = 44 * 50
⇒ CSAᶜᵒⁿᵉ = 2200
∴ The curved surface area of the cone is 2200 cm².
Step-by-step explanation:
Volume of a cone = 9856cm^3
Diameter = 28cm
r = 28cm/2 = 14cm
(i) Volume of a cone = 1/3πr^2h
9856cm^3 = 1/3 x 22/7 x 14 x 14 x h
9856cm^3 x 21/22 x 196cm^2 = h
16cm x 3 = h
48cm = h
(ii) l^2 = h^2 + r^2
l^2 = (14)^2 + (48)^2
l^2 = 196cm^2 + 2304^2
√l^2 = √2500y^2
√l^2 = √2 x 2 x 25 x 25
l = 2 x 25
l = 50cm
(iii) Curved Surface Area of the cone = πrl
= 22/7 x 14cm x 50cm
= 22 x 2cm x 50cm
= 2200cm^2