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A solid metallic right circular cone ...

A solid metallic right circular cone 20 cm high with vertical angle `60o` is cut into two parts at the middle point of its height by a plane parallel to the base. If the frustum, so obtained, be drawn into a wire of diameter `1/(16)\ c m` , find the length of the wire.

Text Solution

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Let OAB be the cone in which `/_AOB=60^(@)`.
Clearly `/_DOE = 30^(@), OE = 10 cm, OF = 20 cm`.
Let ED = r cm and FB = R cm.

`:." "(ED)/(OE)=tan30^(@)rArr(ED)/10=1/sqrt3`
`rArr" "ED=(10xx1/sqrt3)cmrArrr=10/sqrt3cm`
`"and,"(FB)/(OF)tan30^(@)rArr(FB)/20=1/sqrt3`
`rArr" "FB=(20xx1/sqrt3)cmrArrR=20/sqrt3cm.`
Also, EF = 10 cm.
Thus, ABCD is the frustum of the cone in which
`R=20/sqrt3cm, r=10/sqrt3cm" and "h = 10 cm.`
`"Volume of this frustum "=1/3pih(R^(2)+r^(2)+Rr)`
`=1/3xxpixx10(400/3+100/3+200/3)cm^(3)`
`=((pixx10)/3xx700/3)cm^(3)=((700pi)/9)cm^(3)`.
Let the lenfth of the wire be l.
Radius of the wire, `r_(1)=1/32cm`.
`"Volume of the wire"=pir_(1)^(2)=pixx(1/32)^(2)xxl`.
`:." "(700pi)/9=(pil)/(32xx32)rArrl=((700xx32xx32)/9)cm`
`rArr" "l=((7000xx32xx32)/(9xx100))m=(71680/9)m`
= 7964.44m.
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