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SURFACE AREAS

AND VOLUMES

  • Sum based on Cone and Sphere

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Q. 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.

8 cm

Vol. of water flows out =

Vol. of N lead shots

According to Archimedes principle,

Volume of water displaced =

Volume of submerged body

Water flows out

lead shots

5 cm

Sol.

Vol. water flows out =

N

×

Vol. 1 lead shot

N =

Vol. of water flows out

Vol. 1 lead shot

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Q. 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.

Sol.

Vol. water flows out =

1

4

×

Vol. of water in the cone

1

4

×

1

3

πr2h

=

1

4

1

3

× π

× 5

× 5

× 8

×

2

=

2 × 5 × 5 × π

3

∴ Vol. water flows out =

50π

3

cm3

What is the formula to find volume of a cone?

…(i)

1

3

× πr2h

N =

Vol. 1 lead shot

Vol. of water flows out

5 cm

8 cm

=

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Q. 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.

Sol.

Lead shot is in the form of sphere,

Vol. of one lead shot =

4

3

πr3

=

4

3

× π

× (0.5)3

=

0.5π

3

cm3

…(ii)

=

4

3

× π

× 0.125

∴ Vol. of one lead shot

5 cm

8 cm

N =

Vol. of water flows out

Vol. 1 lead shot

What is the formula to find volume of sphere?

4

3

× πr3

Vol. water flows out =

50π

3

cm3

…(i)

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Q. 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.

Sol.

N =

Vol. of water flows out

Vol. 1 lead shot

=

50 π

3

÷

0.5 π

3

0.5 π

3

=

50 π

3

×

=

50

0.5

=

50 × 10

5

10

=

100

∴ Number of lead shots is 100

Vol. water flow out =

50 π

3

…(i)

Vol. of 1 lead =

0.5 π

3

…(ii)

N =

Vol. of water flows out

Vol. 1 lead shot

5 cm

8 cm