An open cylindrical vessel of intemal diameter 7 and height 8 cm stands on a horizontal table. Irside this is placed a solid metallic right circular cone, he diameter of whose base is Cm and height 8 m Find the volume of water required to fill the vessel If the cone is replaced by another cone, whose heigh is 1cm and the radius of whose base is 2 cm, find the drop in the water level.
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An open cylindrical vessel of internal diameter 7 cm and 8 cm stands on a horizontal table. Inside this is placed a solid metallic right circular cone, the diameter of whose base is 3
2
1
cm and height 8cm. Find the volume of water required to fill the vessel. If the one is replaced by another cone, whose height is 1
4
3
cm and the radius of whose base is 2cm, find the drop in the water level.
Hard
Solution
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Let r be the radius of the cylindrical vessel, r=
2
7
.
Let r
1
be the radius of the cone, r
1
=
2
2
7
=
4
7
.
The volume of water is equal to the difference in the volume of cylindrical vessel and the volume of cone.
V
1
=π(
2
7
)
2
×8−
3
1
×π(
4
7
)
2
×8
=
7
22
×(
2
7
)
2
×8−
3
1
×
7
22
×(
4
7
)
2
×8
=
3
847
cm
3
Therefore, the volume of water required to fill the vessel is
3
847
cm
3
.
As the cone is replaced by new cone which has the radius as 2 cm and height as
4
7
cm, so the volume of water with the new cone will be,
V
2
=π(
2
7
)
2
×8−
3
1
×π(2)
2
×
4
7
=
7
22
×(
2
7
)
2
×8−
3
1
×
7
22
×(2)
2
×
4
7
=
3
902
cm
3
Let h be the drop in the water level, then the drop in the volume of the water is,
V=V
2
−V
1
π(
2
7
)
2
×h=
3
902
−
3
847
7
22
×
2
7
×
2
7
×h=
3
55
h=
21
10
cm
Therefore, the drop in the water level is
21
10
cm.