Question

A hollow sphere of internal and external
diameter 4 cm and 8 cm respectively is
melted to form a cone of base diameter
8cm. Find the height and the slant height
of the cone.​

i would really appreciate if u do both of em

Answer 1

Given :-

• Internal Diameter of Sphere = 4cm

• external Diameter of Sphere = 8cm

• Diameter of cone = 8cm

To Find :-

• The Height and slant height of the cone

Formulae used :-

• Volume of hollow sphere = 4/3π(R³ - r³ )

• Volume cone = 1/3πr²h

Now,

→ volume of hollow sphere = volume of cone

→ 4/3π( R³ - r³ ) = 1/3πr²h

→ 4/3π( (4)³ - (2)²) = 1/3π(4)²h

→ 4/3 ( 64 - 8 ) = 1/3 × 16 × h

→ 4/3 × 56 = 16/3h

→ 4/3 × 56 × 3 = 16h

→ 56 × 4 = 16h

→ (56 × 4)/16 = h

→ 56/4 = h

→ 14 = h

Hence, The height of cone is 14cm.

Now,

→ Slant height = √r² + h²

→ l = √(4)² + (14)²

→ l = √16 + 196

→ l = √212

Therefore, The Slant height of the cone is √212.

Answer 2

1]

Solution :-

Let the radius of external be R and internal r respectively.

Volume of sphere :-

$\sf \dfrac{4}{3} \pi R^{3} - \dfrac{4}{3} \pi r^{3}$

Taking 4/3 and π as common

= 4/3π(R³ - r³)

Volume of cone

$\sf \dfrac{1}{3}\pi r^{2}h$

Here,

Volume of sphere = Volume of cone

Therefore,

By putting values

4/3π[ (4)³ - (2)³] = 1/3π(4)²h

$\sf \dfrac{4}{3}\pi \bigg(64 - 8\bigg)=\dfrac{1}{3}\pi 16 \times h$

Cancelling π

$\sf \dfrac{4}{3} \times 56 = \dfrac{1}{3} \times 16h$

 $\sf \dfrac{4}{3} \times \dfrac{3}{1}\times 56 = 16h$

$\sf 4\times 56 = 16h$

$\sf 224 = 16h$

$\sf h = 224\div 16$

h = 14 m

Finding the slant height.

l = √(r)² + (h)²

l = √(4)² + (14)²

l =  √16 + 196

l =  √212

l = 14.5 m

2]

We know that

TSA  = 4πr^2

1386 = 4 $\times$ 22/7  $\times$ r  $\times$ r

$\sf 1386 = \dfrac{88}7 \times r^{2}$

$\sf 1386 \times 7 = 88 r^{2}$

$\sf 9702 = 88r^{2}$

$\sf \dfrac{9702}{88} = r^{2}$

$\sf 110.25 = r^{2}$

$\sf 10.5 = r$

Volume of sphere = Volume of cylinder

$\sf \dfrac{4}{3}\pi r^{3} = \pi r^{2} h$

Volume of sphere = 4/3 × 22/7 × 10.5 × 10.5 × 10.5

Volume = 4851 cm^3

Now

4851 = π  × r  × r  × h

4851 = 22/7  × r² 31.5

4851 = 22 × 4.5r²

4851 = 99r²

4851/99 = r²

49 = r²

7 = r

Diameter =2r

D = 2(7)

D = 14 cm