Question

9. e selid metallic sphere of
diameter 21cm is melted and
recast into a number of smaller
cones, each of diameter & som
and heigh - 3cm. Find the
number of cones so formed.​

Answer 1

Given:

A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones, each of diameter 3.5 cm and height - 3cm.

To find:

The number of cones so formed

Solution:

Finding the volume of the sphere:

The diameter of the solid metallic sphere = 21 cm

So, the radius of the metallic sphere, r = $\frac{21}{2}$ = 10.5 cm

∴ The volume of the solid metallic sphere is,

=  $\frac{4}{3} \pi r^3$

= $\frac{4}{3} \times \frac{22}{7} \times 10.5^3$

= $\bold{4851 \:cm^3}$

Finding the volume of each small cone:

The diameter of the cone = 3.5 cm

So, the radius of the cone, r = 1.75 cm

The height of the cone, h = 3 cm

∴ The volume of each small cone is,

= $\frac{1}{3} \pi r^2h$

= $\frac{1}{3} \times \frac{22}{7} \times 1.75^2 \times 3$

= $\bold{9.625 \:cm^3}$

Finding the no. of cones:

The no. of cones formed after melting and recasting the given sphere is,

= $\frac{Volume \:of\;sphere}{Volume \:of \:each\: cone}$

= $\frac{4851}{ 9.625}$

= $\bold{504}$

Thus, the number of cones so formed is → 504.

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Answer 2

Given :- A solid metallic sphere of diameter 21cm is melted and recast into a number of smaller cones, each of diameter 3.5 cm height 3cm. Find the number of cones so formed. ?

Solution :-

we know that,

• Volume of sphere = (4/3) * π * (radius)³ .
• Volume of cone = (1/3) * π * (radius)² * height .
• Radius = (Diameter/2) .
• when any solid is recast to form another solid , volume remains same.

Let us assume that, total n smaller cones are formed when solid metallic sphere is melted .

then,

→ Volume of sphere = n * (Volume of cone.)

Putting values we get,

→ (4/3) * π * (21/2)³ = (1/3) * π * (3.5/2)² * 3

(1/3) and π will be cancel ,

→ 4 * (21*21*21/8) = n * (12.25/4) * 3

→ 21 * 21 * 21 * 4 = n * 2 * 12.25 * 3

→ 21 * 21 * 42 = 36.75n

→ n = (21 * 21 * 42) / 36.75

→ n = 504 . (Ans.)

Hence, 504 smaller cones are formed.

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