Question

A spherical ball of lead 3 cm in diameter is melted and recast into three spherical
balls. If the diameters of the small balls are 2 cm, 2 cm and p cm, find
Volume of the ball before melting.
Volume of the each spherical ball after melting.
Find the value of p​

Answer 1

$\underline{\underline{\sf{\maltese\:\:Question}}}$

⭐ A spherical ball of lead 3 cm in diameter is melted and recast into three spherical  balls. If the diameters of the small balls are 2 cm, 2 cm and p cm, Find :

• Volume of the ball before melting.

• Volume of the each spherical ball after melting.

• Value of p

$\underline{\underline{\sf{\maltese\:\:Given}}}$

• Diameter of Ball before Melting = 3 cm

• Diameters of the small balls are 2 cm, 2 cm and p cm

$\underline{\underline{\sf{\maltese\:\:To \:Find}}}$

• Volume of the ball before melting.

• Volume of the each spherical ball after melting.

• Value of p

$\underline{\underline{\sf{\maltese\:\:Answer}}}$

• Volume of the ball before Melting = 14.141 cm³

Volume of the each spherical ball after melting :

• Volume of Spherical Ball having diameter 2 cm = 4.190 cm³, Volume of Spherical Ball having diameter p cm = 4/3πr³

• Value of p​ = 2.224

$\underline{\underline{\sf{\maltese\:\:Diagram}}}$

(Given In Attachment)

$\underline{\underline{\sf{\maltese\:\:Calculations}}}$

We need to know about some basic terms before going to Calculations

• Diameter : The center of a circle is the midpoint of its diameter, It divides the circle into two equal parts

• Radius : The distance from the center to the circumference of a circle

Also :

• Diameter = 2 × Radius

• Radius = Diameter/2

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• Volume of the ball before Melting

Volume of Spherical Ball = 4/3πr³

⇒ Volume of Spherical Ball = 4/3 × π × r³

Since Radius = Diameter/2

⇒ Volume of Spherical Ball = 4/3 × π × (Diameter/2)³

⇒ Volume of Spherical Ball = 4/3 × π × (3 cm/2)³

We know (a/b)³ = a³/b³

⇒ Volume of Spherical Ball = 4/3 × π × (3 cm)³/2³

⇒ Volume of Spherical Ball = 4/3 × π × 27 cm³/8

Since π = 22/7

⇒ Volume of Spherical Ball = 4/3 × 22/7 × 27 cm³/8

⇒ Volume of Spherical Ball = 4/3 × 22/7 × 3.375 cm³

⇒ Volume of Spherical Ball = (4 × 22)/(7 × 3) × 3.375 cm³

⇒ Volume of Spherical Ball = 88/21 × 3.375 cm³

⇒ Volume of Spherical Ball = 4.190 × 3.375 cm³

⇒ Volume of Spherical Ball = 14.141 cm³

_____________________________________________

• Volume of the each spherical ball after melting

Volume of Spherical Ball having diameter 2 cm = 4/3πr³

⇒ Volume of Spherical Ball having diameter 2 cm = 4/3 × π × r³

Since Radius = Diameter/2

⇒ Volume of Spherical Ball having diameter 2 cm = 4/3 × π × (Diameter/2)³

⇒ Volume of Spherical Ball having diameter 2 cm = 4/3 × π × (2 cm/2)³

⇒ Volume of Spherical Ball having diameter 2 cm = 4/3 × π × (1 cm)³

⇒ Volume of Spherical Ball having diameter 2 cm = 4/3 × π × 1 cm³

Since π = 22/7

⇒ Volume of Spherical Ball having diameter 2 cm = 4/3 × 22/7 × 1 cm³

⇒ Volume of Spherical Ball having diameter 2 cm = (4 × 22)/(3 × 7) cm³

⇒ Volume of Spherical Ball having diameter 2 cm = 88/21 cm³

⇒ Volume of Spherical Ball having diameter 2 cm = 4.190 cm³

2nd Ball has same Diameter as 1st one, so take before calculations for it

Volume of Spherical Ball having diameter p cm = 4/3πr³

⇒ Volume of Spherical Ball having diameter p c = 4/3 × π × r³

Since Radius = Diameter/2

⇒ Volume of Spherical Ball having diameter p cm = 4/3 × π × (Diameter/2)³

⇒ Volume of Spherical Ball having diameter p cm = 4/3 × π × (p cm/2)³

We know (a/b)³ = a³/b³

⇒ Volume of Spherical Ball having diameter p cm = 4/3 × π × (p³/2³) cm³

⇒ Volume of Spherical Ball having diameter p cm = 4/3 × π × p³/8 cm³

Since π = 22/7

⇒ Volume of Spherical Ball having diameter p cm = 4/3 × 22/7 × p³/8 cm³

⇒ Volume of Spherical Ball having diameter p cm

= (4 × 22)/(3 × 7)  × p³/8 cm³

⇒ Volume of Spherical Ball having diameter p cm = 88/21  × p³/8 cm³

⇒ Volume of Spherical Ball having diameter p cm = (88 × p³)/(21 × 8) cm³

⇒ Volume of Spherical Ball having diameter p cm = 88p³/(21 × 8) cm³

⇒ Volume of Spherical Ball having diameter p cm = 11p³/(21 × 1) cm³

⇒ Volume of Spherical Ball having diameter p cm = 11p³/21 cm³

⇒ Volume of Spherical Ball having diameter p cm = 0.523 p³ cm³

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• Value of p

Volume of the three new sphere = Volume of old sphere

⇒ 4.190 cm³ + 4.190 cm³ + 0.523 p³ = 14.141 cm³

⇒ 8.38 cm³ + 0.523 p³ = 14.141 cm³

⇒ 0.523 p³ = 14.141 cm³ - 8.38 cm³

⇒ 0.523 p³ = 5.761

⇒ 0.523 p³ × 1000 = 5.761 × 1000

⇒ 523 p³ = 5761

⇒ p³ = 5761/523

⇒ p³ = 11.015

⇒ p = 2.224

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Answer 2

Answer:

In this case the volume of the initial spherical ball will be the sum of the volumes of the new three balls.

  Volume of a sphere =3/4πr3

As the diameter of the given sphere is 3cm then its radius =2/3=1.5cm ......... as [radius=diameter/2

The radius of other two balls will be .75cm and 1cm.

Let the radius of the third new ball be Rcm.

Volume of the three new sphere = Volume of old sphere

⇒ 4/3π.75³ + 4/3π.1³ + 4/3π.R³ = 4/3π1.5³

⇒ .75³ + 1³ + R³ =   1.5³

⇒ 0.421875 + 1 + R³= 3.375

⇒ 1.421875 + R³ = 3.375

⇒ R³=1.953125

⇒ R=1.953125 1/3

=> R = 1.25

so, the radius of the third new ball is 1.25.

therefore, the diameter = 2.5cm