Question

The surface area of a solid metallic sphere is 616 cm². It is melted and recast into smaller
spheres of diameter 3.5 cm. How many such spheres can be obtained?
​

Answer 1

Answer:

4πr² = 616 cm²

πr² = 616/4 = 154

r² = 1078/22 = 49

r = √49 = 7cm

V = 4/3 πr³

V = 4/3 × 22/7 × 7 × 7 × 7

V = 4/3 × 154 × 7

V = 4/3 × 1078

V = 4 × 359.3

V = 1436.76 cm³

Small spheres :

v = 4/3 × 22/7 × 3.5/2 × 3.5/2 × 3.5/2

v = 4/3 × 22 × 0.5/2 × 12.25/4

v = 4/3 × 11 × 0.5 × 12.25/4

v = 269.5/12

v = 22.45 cm³

No. of spheres can be made = V / v

=> 1436.76 / 22.45

=> approx. 63.9 or 64 spheres

Answer 2

$\bold{\underline{\underline{\huge{\sf{ANSWER\::}}}}}$

Given:

The surface area of a solid metallic sphere is 616cm². It is melted and recast into smaller spheres of diameter 3.5cm.

To find:

The spheres are can be obtained.

Explanation:

We have,

The surface area of a solid sphere= 616cm²

We know that formula of the surface area of sphere: 4πr²     [sq.units]

→ 4× $\frac{22}{7}$ ×r² = 616cm²

→ 88/7 × r² = 616

→ r² = $\frac{616*7}{88}$

→ r² = $\frac{\cancel{616}*7}{\cancel{88}}$

→ r² = (7× 7)cm

→ r² = 49cm

→ r= √49cm

→ r= 7cm.

Therefore,

We know that volume of the sphere: $\frac{4}{3} \pi r^{3}$  

Volume of the big sphere= $(\frac{4}{3} *\frac{22}{7} *7*7*7)cm^{3}$

Volume of the big sphere= $(\frac{4}{3} *\frac{22}{\cancel{7}} *\cancel{7}*49)cm^{3}$

Volume of the big sphere= $(\frac{88*49}{3} )cm^{3}$

Volume of the big sphere= $\frac{4312}{3} cm^{3}$

Volume of the big sphere= 1437.33cm³

&

Volume of the small sphere:

We have,

• Diameter of small sphere,d= 3.5cm
• Radius of small sphere,r= 3.5/2cm

→ $(\frac{4}{3} *\frac{22}{7} *\frac{3.5}{2} *\frac{3.5}{2} *\frac{3.5}{2} )cm^{3}$

→ $(\frac{\cancel{88}}{21} *\frac{42.875}{\cancel{8}} )cm^{3}$

→ $(\frac{11}{21} *42.875)cm^{3}$

→ $\cancel{(\frac{471.625}{21} )}cm^{3}$

→ 22.45cm³.

• ∴ The number of smaller sphere:

⇒ $\frac{Volume\:of\:big\:sphere}{Volume\:of\:small\;sphere}$

⇒ $\cancel{\frac{1437.33cm^{3} }{22.45cm^{3} }}$

⇒ 64.02 spheres

Thus,

The sphere can be obtained is 64 spheres .  [approximately]