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Question 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 116 cm, find the length of the wire.


Solution

Volume of frustum will be equal to the volume of wire and by using this relation we can calculate the length of the wire.


In the given figure; AO = 20 cm and hence height of frustum PO = 10 cm
In triangle AOC we have angle CAO = 30 (halft of vertical angle of cone BAC)
Therefore;
tan 30=OCAO
Or,13=OC20
Or,OC=203
Using similarity criteria in triangles AOC and ADE it can be shown that DE=103 (because DE bisects the cone through its height)
Similarly, PO = 10 cm
Volume of frustum can be calculated as follows:
V=13πh(r21+r22+r1r2)
=13π×π10[(203)2+(103)2+203×103]
=13π×10(4003+1003+2003)
=70009π cm3
Volume of cylinder = πr2h
Or,π×(132)2×h
=70009π
Or,h=70009×1024
=796444.44 cm

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