man
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 7 cm, find the length of the wire.
Answers
Answer:
Let OCD be the metallic cone
and ABCD be the required frustum
Here frustum is drawn as wire
Volume of frustum ABCD = Volume of cylindrical wire
Volume of frustum ABCD
Volume of frustum = πh / 3 ( r1² + r2² + r1.r2 )
Given OP = 20 cm
As cone is cut at the middle
OQ = QP = 10 cm
angle QOB = 60°
Height of frustum = QP = 10 cm
r1 = ?
r2 = ?
r1 = PD
r2 = QB
♦️In right angle triangle POD
tan O = PD / OP
tan 30° = PD / OP
1 / √3 = r1 / 20
r1 = 20 / √3 cm
♦️In right angle triangle QOB
tan O = QB / OQ
tan 30° = QB / OQ
1 / √3 = r2 / 10
r2 = 10 / √3 cm
♦️♦️Volume of frustum ABCD = πh / 3 ( r1² + r2² + r1.r2 )
→ π . 10 / 3 ( (20/√3)² + (10/√3)² + 20/√3 . 10/√3 )
→ 10π / 3 ( 400 / 3 + 100 / 3 + 200 / 3 )
→ 10π / 3 ( 700 / 3 )
→ 7000π / 9 cm²
♦️♦️Volume of cylindrical wire :-
Given
diameter = 1 / 16 cm
Radius = 1/16 / 2 = 1 / 16 . 2 = 1 / 32 cm
let ,
length of the wire = height of the cylinder = h cm
♦️Volume of the wire = πrh
→ π ( 1/32 )² h
→ πh / 32 . 32
⭐ Volume of the frustum = Volume of wire
7000π / 9 = πh / 32 . 32
7000π × 32 × 32 / 9 × π = h
h = 7000π × 32 × 32 / 9 × π
h = 796444.44 cm
h = 796444.44 / 100 cm
h = 7964.4 m
:. Length of the wire = h = 7964.4 m .
Answer:
Step-by-step explanation:
