A solid cone of height 12cm and base radius 6cm has the top 4cm removed as shown in the fig. Find the whole surface area of the remaining frustum of the cone.
Answer:349.888 cm2 , just need the process
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Secondary SchoolMath 5 points
A solid cone of height 12 cm and base radius 6 cm has the top 4 cm removed. Find the whole surface area of remaining figure.
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abhi178
abhi178
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A cone of radius CD and height AD as shown in figure is cut from the top of 4cm at point E.
now, AD = 12cm , CD = 6cm , AE = 4cm
here it is clear that ∆ABE ~ ∆ACD
so, AE/AD = BE/CD
4cm/12cm = BE/6cm
BE = 2 cm { it is the radius of small circular part , Let r }
now, whole surface area of remaining part of cone = lateral surface area of frustum + area of above circular part + area of below circular part
= πl(R + r) + πr² + π R²
where, l = √{h² + (R - r)²}
here, h = 12cm - 4cm = 8 cm
so, l = √{8² + (6-2)²} = √{64 + 16} = 4√5cm
now, whole surface area = π × 4√5 × (6 +2) + π × (6)² + π × (2)²
= 32√5π + 36π + 4π cm²
= (32√5 + 40)π cm²
= (32 × 2.236 + 40) × 22/7 cm²
= 350.59 cm²
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nikitasingh79
nikitasingh79 Genius
Given:
AB= 4 cm, AC= 12 cm, CD = 6 cm
In Δ ABE and Δ ACD,
BE || CD
∠AEB= ∠ADC [each 90°]
∠ABE= ∠ACD [ corresponding angles]
Δ ABE ∼ Δ ACD [By AA Similarity]
AB/AC = BE/CD
[Corresponding sides of a similar triangles are proportional]
4/12 = BE /6
1/3 = BE/6
1 = BE/2
BE = 2
In ∆ACD
AD² = AC² + CD²
AD² = 12² + 6²
AD² = 144 + 36
AD²= 180
AD = √180 = √36×5 = 6√5 =6×2.236
Slant height of bigger cone AD = 13.416 cm
Total surface area of bigger cone with radius 6 cm = πr(l + r)
= π×6(6 + 13.416)
= π×6×19.416= π(116.496) cm²
Slant height of smaller cone (l) =√h²+r² √(AB²+BE² )
l = √(4²+ 2²)
l = √(16 + 4)
l = √20 =√4×5=2×2.236
l = 4.472 cm
Curved surface area of smaller cone of height 4 cm and radius 2 cm = πrl
= π×2×4.472 = π(8.944) cm
Total surface area of the remaining cone = Total surface area of bigger cone - curved surface area of smaller cone + area of base of smaller cone
= π(116.496) - π(8.944) + πr²
= π(116.496) - π(8.944) + π(2)²
= π(116.496 - 8.944 +4)
= π(107.552 +4) = π (111.552) cm
= 22/7(111.552)= 2,454.144 /7 = 350.59 cm²
Hence, the Total surface area of the remaining cone = 350.59 cm²