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प्रश्न
Choose the correct answer of the following question:
A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is
विकल्प
1 : 2
1 : 4
1 : 6
1 : 8
उत्तर
Let the radii of the smaller and given cones be r and R, respectively; and their heights b and H respectively
We have,
H = 2h .........(i)
In ΔAQD and ΔAPC,
∠QAD = ∠PAC (common angle)
∠AQD = ∠APC = 90°
So, by AA citeria
Δ AQD ˜ Δ APC
`=> "AQ"/"AP" = "QD"/"PC"`
`=> "h"/"H" = "r"/"R"`
`=> "h"/"2h" = "r"/"R"`
`=>1/2 = "r"/"R"`
⇒ R = 2r ........(ii)
Now,
The ratio of the smaller cone to the whole cone`="Volume of the smaller cone"/"Volume of the whole cone"`
`= ((1/3pi"r"^2"h"))/((1/3pi"R"^2H))`
`=("r"/"R")^2 xx ("h"/"H")`
`=("r"/"2r")^2xx("h"/"2h")` [Using (i) and (ii)]
`=(1/2)^2 xx (1/2)`
`=1/8`
= 1 : 8
Hence, the correct answer is option (d).
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