A cone of radius 4 cm is divided into two parts by drawing a plane through the mid point of its
axis and parallel to its base. Compare the volumes of the two parts.
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Answers
A cone of radius 4 cm is divided into 2 parts by drawing a plane through mid point of its axis parallel to it's base.
Please see the attachment for figure.
We got two parts Top and bottom
Top part is shape of cone whose height is h and radius r
Volume of cone =\frac{1}{3}\times \pi\times r^2\times h=
3
1
×π×r
2
×h
Volume of Top part =\frac{1}{3}\times \pi\times r^2\times h=
3
1
×π×r
2
×h
Bottom part is shape of frustum whose height is h and top radius r and bottom radius R.
Using similar theorem property, ΔOAD ≈ΔOBC
\therefore \dfrac{OA}{OB}=\dfrac{AD}{BC}∴
\dfrac{h}{2h}=\dfrac{r}{R}
2h
h
R=2r
Volume of bottom part =\frac{1}{3}\times \pi\times h(R^2+r^2+Rr)= 31
×π×h(R 2 +r 2 +Rr)
Volume of bottom part =\frac{1}{3}\times \pi\times h(4r^2+r^2+2r^2)=\frac{1}{3}\times \pi\times h\times 7r^2= 1
×π×h(4r 2
+r 2
+2r )=3
1 ×π×h×7r 2
Now we find the ratio of both volume \dfrac{\frac{1}{3}\times \pi\times r^2\times h}{\frac{1}{3}\times \pi\times h\times 7r^2}
3
1
×π×h×7r
2
3
1
×π×r
2
×h
Ratio = 1:7
Hence, volume of bottom part is 7 times the volume of top part.
Answer:
The \: radius \: of \: upper \: cone (R _{1}) = \frac{4}{2 } = 2Theradiusofuppercone(R
1
)=
2
4
=2
Bottom radius 4cm
Volume \: of \: upper \: cone (V _{1}) = \frac{1}{3)} \pi r^{2} hVolumeofuppercone(V
1
)=
3)
1
πr
2
h
= \frac{1}{3} \pi \times ({2}^{2}) \times ( \frac{h}{2} )=
3
1
π×(2
2
)×(
2
h
)
Vol \: of \: bottom \: frustum (V_{2}) = \frac{\pi h}{3} (R \frac{2}{1} R \frac{2}{2} + R _{1}R _{2})Volofbottomfrustum(V
2
)=
3
πh
(R
1
2
R
2
2
+R
1
R
2
)
= \frac{\pi( \frac{h}{2}) }{3} (28)=
3
π(
2
h
)
(28)
Vol \: of \: bottom \: frustum (V _{2}) = \frac{\pi h}{6} (28)Volofbottomfrustum(V
2
)=
6
πh
(28)
\frac{V _{1} }{ V_{2} } = \frac{2\pi h}{3} \times \frac{6}{\pi h \times 28}
V
2
V
1
=
3
2πh
×
πh×28
6
\frac{V _{1}}{V _{2}} = \frac{2\pi h}{3} \times \frac{6}{\pi h \times 28}
V
2
V
1
=
3
2πh
×
πh×28
6
\frac{V_{1} }{ V_{2} } = \frac{1}{7}
V
2
V
1
=
7
1
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