Question

$can \: ya \: all \: please \: help$
A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered into the water such that when it touches the sides, it is just immersed.What fraction of water overflows?

Answer 1

Answer:

3:8

Step-by-step explanation:

Let AM = 8 cm, MB = 6 cm.

From figure:

(i)

In ΔAMB,

AB² = AM² + MB²

      = 8² + 6²

      = 100 cm.

AB = 10 cm.

∴ CB is tangent at M and AB is tangent to P.

⇒ PB = MB = 6

⇒ AP = AB - PB

         = 10 - 6

         = 4 cm.

Let r be the radius of the circle,

OP = OM = r.

∴ AO = AM - OM

         = 8 - r

(ii)

In ΔOAP,

OA² = AP² + OP²

⇒ (8 - r)² = 4² + r²

⇒ 64 + r² - 16r = 16 + r²

⇒ 48 = 16r

⇒ r = 3 cm.

∴ Volume of the sphere = Volume of water which overflows:

= (4/3) πr³

= (4/3) * π * (3)³

= 36π cm³.

∴ Volume of water in the cone before immersing the sphere = Volume of cone

= (1/3) * π * (6)² * 8

= 96 π cm³.

∴ Fraction of water overflows = 36π/96π

                                                 = 3/8

                                                 = 3 : 8.

Therefore, Fraction of water overflows = 3 : 8.

Hope it helps!