A metallic right circular cone 20 cm high and whose vertical angle is 60° is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter 1/6 cm find the length of the wire.

In ΔAEG,
EG/AG = tan 30°
⇒ EG/10 = 1/√3
⇒ EG = 10/√3
In ΔABD,
BD/AD = tan 30°
⇒ BD/20 = 1/√3
⇒ BD = 20√3
Radius of upper part of frustum (r₁) = 10/√3
Radius of lower part of frustum (r₂) = 20/√3
Height of frustum (h) = 10 cm
Volume of frustum = 1/3πh(r₁² + r₂² + r₁r₂)
= 1/3π × 10 × [(10/√3)² + (20/√3)² + 10/√3 × 20/√3]
= 1/3 × 22/7 × 10 × (100/3 + 400/3 + 200/3)
= 1/3 × 22/7 × 10 × (700/3) = 22000/9 cm³
Radius of wire (r) = 1/2 × 1/16 = 1/32 cm
Let the lenght of wire = l
Volume of wire = area of cross- section of wire × length of wire
= (πr²) (l)
= π(1/32)²× l
Volume of frustum = Volume of wire
⇒ 22000/9 = π(1/32)² × l
⇒ 22000/9 = 22 /7 × 1/1024 × l
⇒ l = 22000/9 × 7/22 × 1024
= 796444.44 cm
7964. 44 m
Hence, the length of wire is 7964.44 m.