NCERT Solutions for Class 10 Maths

Surface areas and Volumes,

NCERT Books for Session 2022-2023

CBSE Board and UP Board Others state Board

EXERCISE 13.4

Page No:257

Questions No:5

# 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.

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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.