Q. 194.1( 29 Votes )
A solid 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 1/12cm, find the length of the wire.
Answer :
Let ‘R’ be the radius of the base of the cone which is also the base of frustum i.e. lower circular end as shown in the figure
DE = R
Let ‘r’ be the radius of the upper circular end of frustum which we get after cutting the cone
BC = r
The height of the cone is 20 cm and we had cut the cone at midpoint therefore height of the frustum so obtained is 10 cm
Vertical angle as shown in the figure is 60˚
Now a wire of diameter 1/12 (i.e. radius 1/24) is made out of the frustum let ‘l’ be the length of the wire
As we are using the full frustum to make wire therefore volumes of both the frustum and the wire must be equal.
∴ Volume of frustum = volume of wire made …(i)
Consider ΔABC
∠BAC = 30˚ ; AB = 10 cm ; BC = r
∴ r = 10/√3 cm
Consider ΔADE
∠DAE = 30˚ ; AD = 20 cm ; DE = R
∴ R = 20/√3 cm
Now using equation (i)
∴ 7000/9 = l/576
∴ 777.778 = l/576
∴ l = 448000 cm
Length of wire = 448000 cm
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