The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If it's volume be 1/27 of given cone , at what height above the base , the section has been made?
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Assume that big cone of radius R and H.
Assume that small cone of height h is cut off from the top of this cone whose base is parallel to the big cone.
And the radius of the cone that cut off from the original cone be r.
Given H = 30 cm.
In ΔAPC and ΔAQE
PC || QE
ΔAPC ~ ΔAQE
AP / AQ = PC / QE
h / H = r / R -------------(1)
Given that Volume of cone ABC = (1 / 27) Volume of cone ADE
Volume of cone ABC / Volume of cone ADE = 1 / 27
(1/3)pr2h / (1/3)pR2H = 1 / 27
(r/R)2 x (h/H) = 1 / 27
(h / H)2 x ( h / H) = (1 /3)3 [ from (1) ]
( h / H)3 = (1 /3)3
( h / H) = (1 /3)
h = ( 1/3) x H
h = ( 1/3) x 30 = 10 cm
From the figure PQ = H – h = 30 – 10 = 20 cm.
Hence, the section of the cone is made at a height of 20 cm above the base.
Assume that small cone of height h is cut off from the top of this cone whose base is parallel to the big cone.
And the radius of the cone that cut off from the original cone be r.
Given H = 30 cm.
In ΔAPC and ΔAQE
PC || QE
ΔAPC ~ ΔAQE
AP / AQ = PC / QE
h / H = r / R -------------(1)
Given that Volume of cone ABC = (1 / 27) Volume of cone ADE
Volume of cone ABC / Volume of cone ADE = 1 / 27
(1/3)pr2h / (1/3)pR2H = 1 / 27
(r/R)2 x (h/H) = 1 / 27
(h / H)2 x ( h / H) = (1 /3)3 [ from (1) ]
( h / H)3 = (1 /3)3
( h / H) = (1 /3)
h = ( 1/3) x H
h = ( 1/3) x 30 = 10 cm
From the figure PQ = H – h = 30 – 10 = 20 cm.
Hence, the section of the cone is made at a height of 20 cm above the base.
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