Question

A hollow metallic sphere with external diameter 6cm and internal diameter 4cm is melted and moulded into a cone having base radius 2cm, Then the height of the cone is

Answer 1

GIVEN:

• External diameter of the hollow sphere = 6 cm
• Internal diameter of the hollow sphere = 4 cm
• Radius of cone = 2 cm

TO FIND:

• What is the height of the cone ?

SOLUTION:

We have given that the external diameter of the hollow sphere = 6 cm

$\bf{ Radius = \dfrac{Diameter}{2} = \cancel\dfrac{6}{2}}$

$\bf{ External \: radius = 3 \: cm}$

The internal diameter of the hollow sphere = 4 cm

$\bf{ Internal \: radius = \cancel\dfrac{4}{2} }$

$\bf{ Internal \: radius = 2 \: cm}$

To find the volume of hollow sphere, we use the formula:-

$\large\bf{\star \: Volume = \dfrac{4}{3} \pi (R^3 - r^3) \: \star}$

To find the volume of the cone, we use the formula:-

$\large\bf{\star \: Volume = \dfrac{1}{3} \pi r^2 h \: \star}$

Let the height of the cone be 'h' cm

According to question:-

$\bf{ \dfrac{4}{\cancel{3}} \cancel{\pi} (R^3 - r^3) = \dfrac{1}{\cancel{3}} \cancel{\pi} r^2 h }$

$\bf{\dashrightarrow 4(R^3-r^3) = r^2 h }$

On putting the given values in the formula, we get

$\rm{\dashrightarrow 4(3^2-2^2) = 2^2 \times h }$

$\rm{\dashrightarrow 4(9-4) = 4 \times h }$

$\rm{\dashrightarrow 4 \times 5 = 4 \times h }$

$\rm{\dashrightarrow 20 = 4h }$

$\rm{\dashrightarrow h = \cancel\dfrac{20}{4} }$

$\bf{\dashrightarrow \star \: h = 5 \: cm \: \star }$

❝ Hence, the height of the cone is 5 cm ❞

______________________

Answer 2

SOLUTION :

Given : Internal diameter of hollow sphere(d)= 4 cm.

Internal radius of hollow sphere (r) = 4/2= 2 cm

external diameter of hollow sphere (D) = 8 cm.

external radius of hollow sphere( R )= 8/2= 4 cm.

Volume of the Hollow sphere = 4/3π(R³ - r³)

Volume of the Hollow sphere = 4/3π(4³ - 2³)

Volume of the Hollow sphere = 4/3π(64 - 8)

Volume of the Hollow sphere = 4/3π(56) cm³

Diameter of the cone(d1) = 8 cm

radius of the cone( r1)= 8/2 = 4 cm

Let the height of the cone be h cm.

Volume of the cone = ⅓ πr1²h

= ⅓ π × 4² × h = 16πh/3

Volume of the cone = Volume of the hollow sphere

16πh/3 = 4/3π(56)

16h = 4 ×56

h = (4 × 56)/16

h = 56/4 = 14 cm

Hence, the height of the cone is 14 cm.