29. A conical vessel of radius 6 cm and height 8 cm is completely fill

1.	29. A conical vessel of radius 6 cm and height 8 cm is completely filled with water. Asphere is lowered into the water and its size is such that it touches the sides it isjust immersed. What fraction of water overflows?
Answer» Radius (R) of conical vessel = 6 cm Height (H) of conical vessel = 8 cm volume of conical vessel (Vc) = 1/3πr^2h= 1/3π × 36 × 8= 96πLet the radius of the sphere be r cm. In right ΔPO'R, by Pythagoras theorem: L = √(64 + 36) L = 10 cm Hence sin = O'P / PR = 6/10 = 3/5 In right triangle MRO Sin = OM /OR = r / OR 3/5 = r / (8 - r) ⇒ 24 – 3r = 5r ⇒ 8r = 24 ⇒ r = 3 cm ∴ Volume of sphere (Vs) Now, Volume of the water = Volume of cone (Vc) = 96 π cm3 Clearly, Volume of the water that flows out of cone is same as the volume of the sphere i.e. Vs. ∴ Fraction of the water that flows out Vs/Vc = 36π / 96π = 3/8Vs : Vc = 3 : 8

29. A conical vessel of radius 6 cm and height 8 cm is completely filled with water. Asphere is lowered into the water and its size is such that it touches the sides it isjust immersed. What fraction of water overflows?

Answer»

Radius (R) of conical vessel = 6 cm

Height (H) of conical vessel = 8 cm

volume of conical vessel (Vc)

= 1/3πr^2h= 1/3π × 36 × 8= 96πLet the radius of the sphere be r cm.

In right ΔPO'R, by Pythagoras theorem:

L = √(64 + 36)

L = 10 cm

Hence sin = O'P / PR = 6/10 = 3/5

In right triangle MRO

Sin = OM /OR = r / OR

3/5 = r / (8 - r)

⇒ 24 – 3r = 5r

⇒ 8r = 24

⇒ r = 3 cm

∴ Volume of sphere (Vs)

Now,

Volume of the water = Volume of cone (Vc) = 96 π cm3

Clearly, Volume of the water that flows out of cone is same as the volume of the sphere i.e. Vs.

∴ Fraction of the water that flows out

Vs/Vc = 36π / 96π = 3/8Vs : Vc = 3 : 8