Question

a spherical ball of radius 3 cm is melted and recast into three spherical balls the radii of the two of the balls are 1.5 cm and 2 cm respectively determine the diameter of the third ball ​

Answer 1

Given that:

The radius of spherical ball = 3 cm

The radii of the ball are 1.5 cm and 2cm

To find:

The diameter of third ball.

$\boxed{Volume \:of \:spherical\: ball = 4/3πr3}$

= 4/3π × 3 × 3 × 3

= 36π cm3

Volume of spherical ball = Total volume of three small spherical ball.

Let the radius of third ball = r

$\huge{\sf{\boxed{\boxed{Solution:}}}}$

36π = 4/3π × (3/2)3 + 4/3π × (2)3 + 4/3 πr3

36π = 4/3π × 27/8 + 4/3π × 8 + 4/3

36π = 4/3π(27/8 + 8 + r3)

(36π × 3)/4π = 27/8 + 8 + r3

Therefore,

27 = (27 + 64)/8 + r3

27 = 91/8 + r3

27 – 91/8 = r3

(216 – 91)/8 = r3

125/8 = r3

r = 5/2 cm

The diameter of the third ball = 2r = 2 × 5/2

= 5 cm

Hence

The diameter of the third ball is 5 cm.

Answer 2

Given:

Radius of spherical ball is 3 cm.

Radii of new spherical balls are 1.5 cm and 2 cm.

To Find:

Radius of third spherical ball ?

Solution :

Let the radius of third spherical ball be x cm.

If something is melted and recasted into another thing then their volumes will be equal. In short

Volume of 1st thing = Volume of second one.

➯ Let's see here

Volume of big spherical ball will be equal to the sum of volumes of that three small spherical balls.

As we know that

★ Volume of Sphere = $\blue{\dfrac{4}{3}πr³}$ ★

[ Taking big spherical ball ]

Radius = 3 cm

⟹ Volume = 4/3 × π × (3)³

⟹ 4π/3 × 27

Volume we got = 4π/3 × 27 cm³

[ Taking 3 small spherical balls ]

Radius of first ball (r¹) = 1.5 cm

For second (R) = 2 cm

For third (x) = x cm

Volume = 4/3 × π( sum cubes of radii)

⟹ Volume = 4/3 × π(1.5³ + 2³ + r³)

⟹ 4π/3 (3.375 + 8 + x³)

⟹ 4π/3 ( 11.375 + x³)

Volume we got = 4π/3 (11.375 + x³) cm³

A/q

First volume = Second volume

➮ 4π/3 × 27 = 4π/3 (11.375 + x³)

➮ 27 = 11.375 + x³

➮ 27 – 11.375 = x³

➮ 15.625 = x³

➮ 15625/1000 = x³

➮ 3125/200 = 625/40 = 125/8 = x³

➮ ³√125/8 = x³

➮ 5/2 = x²

➮ 2.5 cm = x

Hence, the measure of radius of third spherical ball is 2.5 cm.

so,Diameter of the third ball is=2×2.5=5cm

$\tt{\underline{\underbrace{\mathcal\color{red}{Answer}}}}$

$\tt{\underline{\underline{\mathcal\color{blue}{Diameter ~of~ the ~third ~ball ~is~ 5cm.}}}}$