Ex 12.2 Q5 - A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.
Solution:-Given,
*For conical vessel:-
Height = h = 8 cm
Radius = r = 5 cm
*For spherical lead shots:-
Radius = R = 0.5 cm
As the water is filled up to the brim in the conical vessel,
∴ Volume of water in the vessel
= Volume of the conical vessel
➙ Volume of conical vessel
=
1
/
3
πr²h =
1
/
3
π × (5)² × 8 =
1
/
3
π × 25 × 8 =
200π
/
3
cm³➙ Volume of a spherical lead shot
=
4
/
3
πR³ =
4
/
3
π × (0.5)³ =
4
/
3
π × 0.125 =
0.5π
/
3
cm³➙ Given that, on dropping certain number of spherical lead shots into the vessel, one-fourth of the water flows out,
Let,
n number of lead shots are dropped in the vessel,
∴ Volume of n spherical lead shots
=
1
/
4
× Volume of water in the vessel ∴ n × Volume of a lead shot
=
1
/
4
× Volume of the conical vessel ∴ n ×
0.5π
/
3
=
1
/
4
×
200π
/
3
∴ n × 0.5 =
200
/
4
∴ n =
200
/
4 × 0.5
∴ n =
200
/
2
∴ n = 100
Hence,
100 spherical lead shots are dropped in the vessel.
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Try This..
✒ How many spherical balls of radius 1 cm can be made by melting a cone of radius 4 cm and height 9 cm?
✒ Lead spheres of radius 0.5 cm are dropped into a cylindrical vessel of radius 3 cm containing water. Find the number of lead spheres required to raise the water level by 2 cm.
✒ A hemispherical bowl of radius 7 cm is filled with milk. How many cones of radius 7 cm and height 6 cm can be filled with this milk?
❌ Common Mistakes
Wrong Condition:▸ Forgetting to take one-fourth of the cone’s volume instead of using the full volume.Incorrect radius:▸ Always double-check if you’re using the right radius—0.5 cm for lead shots here.Mixing up formulas:▸ Use cone formula for the vessel and sphere formula for lead shots.Not simplifying properly:▸ Always cancel π when it’s on both sides to avoid unnecessary calculations.
Queries Solved:-
Class 10 Ex 12.2
Ex 12.2 Q5 Class 10
Class 10 Ex 12.2 Q5
Class 10 Chap 12 Ex 12.2 Q5
Class 10 Surface Areas And Volumes
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