⭐ 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.
- Diameter of Ball before Melting = 3 cm
- Diameters of the small balls are 2 cm, 2 cm and p cm
- Volume of the ball before melting.
- Volume of the each spherical ball after melting.
- 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
(Given In Attachment)
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 :
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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
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- 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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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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