A metallic right circular cone 45cm high and whose Vertical angle 60° cut down into two parts in the ratio 1 : 2 from the vertex of the cone by a plane parallel to its base . If frustum so obtained be drawn into a wire of diameter 1 cm , find the length of the wire
Answers
Given the Vertical Angle of the Cone as : 60°
⇒ Semi Vertical Angle of the Cone will be : 30°
We know the Vertical height divides the Right Circular Cone into Two halves.
Imagining One Half of this Divided Cone by Vertical Height , We can Notice that Radius of the Cone is the Opposite Side to 30° and the Adjacent Side will be Vertical height of 45cm
⇒ Tan30° = (Radius)/45
⇒ Radius of the Cone = 45/√3 = 15√3 cm
We know that the Volume of a Cone is Given by : (πr²h) × 1/3
⇒ Volume of the Given Cone = 22/7 × 15√3 × 15√3 × 45 × 1/3 = 31808.6 cm³
Given that : Cone is Cut down by Someone into two parts in the Ratio 1 : 2
⇒ By this Cutting , Original Cone was Divided into a Small Cone of 15cm with Semi Vertical Angle of 30° and Frustum of 30 cm
Let us Find the Volume of this Small Cone and Subtract it from the Original Cone Volume so that we may end up with the Frustum Volume.
For Small Cone :
Tan30° = (Radius)/15
⇒ Radius of Small Cone = 5√3
Volume of the Small Cone = 22/7 × 5√3 × 5√3 × 15 × 1/3 = 1178.09 cm³
Volume of the Frustum = Volume of the Original Cone - Volume of the Small Cone.
Volume of the Frustum = 31808.6 - 1178.09 = 30630.51 cm³
Given this Frustum is made into Wire of 1cm Diameter (radius = 0.5cm)
⇒ Volume of Frustum = Volume of Wire
We know that Wire is in the shape of Cylinder whose Volume is = πr²l
⇒ Volume of Wire = 22/7 × 0.5 × 0.5 × Length
⇒ 22/7 × 0.5 × 0.5 × Length = 30630.51 cm³
⇒ 0.785 × Length = 30630.51
⇒ Length = (30630.51)/0.785 = 39019.75 cm
⇒ Length of Wire = 390.1975 meter
Answer:
Refer the attachments below :)
Step-by-step explanation:
