Question

Please answer it ❣️❣️​

Answer 1

Question :

A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A  sphere is lowered into the water and its size is such that it just immersed in the vessel as shown in the figure. Find the amount of water displaced by the sphere. Keep the answer in terms of π. Also find the fraction of volume of water displaced that flows out of the vessel.

Answer :

Let's draw the rough figure ( Refer to attachment )

Radius of the base of the cone ( r ) = 6 cm

Height of the cone ( h ) = 8 cm

Firstly let's find the volume of cone and then sphere

We know that

Volume of the cone = 1/3 × πr²h cu.units

                                 = 1/3 × π × 6² × 8

                                = 36 × 8 × π / 3

                               = 12 × 8 × π

                               = 96π cm³

Now, to find out the volume of the sphere we need radius of the sphere , but we have no radius of sphere given in the question. So, now we will find the radius of the sphere.

For that join O and B to get a ΔABC

Let the radius of the sphere be ' r ' cm

Consider Δ ABC

Slant height of the cone ( l ) = √( r² + h² )

⇒ l = √( 6² + 8² )

⇒ l = √( 36 + 64 )

⇒ l = √100

⇒ l = BC = 10 cm

∴ sin θ = Perpendicular / Hypotenuse = AC / BC = 6/10 = 3/5

Consider ΔBDO

OD ⊥ BC   ( Tangent to a circle is perpendicular to the radius at the point of contact )

⇒ ∠BDO = 90°

∴ ΔBDO is a Right angled triangle

⇒ sin θ = Perpendicular / Hypotenuse = DO / BD = DO /  ( AB - OA ) = r / ( 8 - r )

⇒ 3/5 = r / ( 8 - r )

⇒ 3( 8 - r ) = 5r

⇒ 24 - 3r = 5r

⇒ 24 = 5r + 3r

⇒ 24 = 8r

⇒ r = 24/8 = 3 cm

∴ Radius of the sphere is 3 cm

We know that

Volume of the sphere = 4/3 × πr³ cu.units

                                     = 4/3 × π × 3³

                                     = 4 × 9 × π

                                    = 36π cm³

Amount of water displaced = Volume of the sphere = 36π

Therefore the amount of water displaced by the sphere is 36π cm³.

Fraction of volume of water that flows out of the vessel = Volume of the displaced water / Volume of water before the lowering the sphere = 36π / 96π = 3/8

Therefore the fraction of water that flows of the vessel is 3/8.

Answer 2

Step-by-step explanation:

_______________________________

$\bf \underline{Question}$

A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A  sphere is lowered into the water and its size is such that it just immersed in the vessel as shown in the figure. Find the amount of water displaced by the sphere. Keep the answer in terms of π. Also find the fraction of volume of water displaced that flows out of the vessel.

_______________________________

$\bf \underline{Given}$

• A conical vessel of radius 6 cm
• height 8 cm is completely filled with water.
• . A  sphere is lowered into the water.

_______________________________

$\bf \underline{To\:Find}$

• Find the amount of water displaced by the sphere. Keep the answer in terms of π. Also find the fraction of volume of water displaced that flows out of the vessel.

_______________________________

$\bf \underline{Solve}$

From the similarly:-

8 -r/ 10 = r/6

On cross Multiplying:-

48-6r=10r

48=10r+6r

48=16r

r= 3

Fraction of water overflow:-

putting values:-

(4/3π3^3)/(1/3π6^2(8))

R = 3/8

Hence answer is 3/8