Question

The surface area of a solid metallic sphere is 5024 cm2 it is melted and recast into right circular cones of radius 2.5cm and height 8cm calculate the number of cones recast (take pie as 3.14j

Answer 1

Solution :-

Surface area of a solid metallic sphere = 5024 cm²

⇒ 4πr² = 5024

⇒ πr² = 1256

⇒ 3.14 * r² = 1256

⇒ r² = 1256/3.14

⇒ r² = 400

⇒ r² = 20²

⇒ r = 20

Radius of the sphere ( r ) = 20 cm

Solid metallic sphere is melted and recast into right circular cones of radius 2.5cm and height 8 cm.

Radius of cone ( r' ) = 2.5 cm

Height of the cone ( h ) = 8 cm

Let the number of cones formed be n

According to the question :-

⇒ n * Volume of cone = Volume of solid metallic sphere

⇒ n * 1/3 π( r' )² h = 4/3 * πr³

⇒ n * ( r' )²h = 4r³

⇒ n * (2.5)² * 8 = 4 * (20)³

⇒ 6.25n * 2 = 8000

⇒ 6.25n = 4000

⇒ n = 4000/6.25

⇒ n = 400000/625

⇒ n = 640

Therefore, the number cones recasted are 640.

Answer 2

$\huge\star{\pink{\underline{\mathfrak{Answer :-}}}}$

640 cones.

$\huge\star{\pink{\underline{\mathfrak{Explanation :-}}}}$

The surface area of the sphere given = 5024 cm^2.

$\therefore {\leadsto {4 \pi {r}^{2} = 5024 {cm}^{2}}}$

$\leadsto {4×3.14×{r}^{2} = 5024 {cm}^{2}}$

$\leadsto {12.56×{r}^{2} = 5024 {cm}^{2}}$

$\leadsto {{r}^{2} = \dfrac {5024}{12.56}}$

$\leadsto {{r}^{2} = \dfrac {502400}{1256}}$

$\leadsto {{r}^{2} = 400}$

$\leadsto {r = \sqrt{400}}$

$\leadsto \bold {r = 20 cm}$

So, the radius of the sphere = r1 = 20 cm.

Now, it is given that the sphere is melted and recast into small cones of radius(r2) 2.5 cm and height 8 cm.

$\therefore \leadsto {Volume\:of\: the \:sphere = x \:( volume \:of \:one\: cone)}$

(Let x be the numberof cones formed)

$\leadsto {\dfrac {4}{3} \pi {(r1)}^{3} = x (\dfrac{1}{3} \pi {(r2)}^{2} h)}$

$\leadsto {\dfrac {4}{3} \times 3.14 \times {20}^{3} = x(\dfrac {1}{3} \times 3.14 \times {2.5}^{2} \times 8)}$

$\leadsto {x = \dfrac {\dfrac {4}{\cancel {3}} \times \cancel {3.14} \times {20}^{3}}{\dfrac{1}{\cancel {3}} \times \cancel {3.14} \times{2.5}^{2} \times 8}}$

$\leadsto{ x =\dfrac{4 \times{20}^{3}}{8 \times{2.5}^{2}}}$

$\leadsto {x = \dfrac {32,000}{50}}$

$\leadsto\bold { x = 640 \: cones}$

$\therefore{The\:total\:number\:of\:cones\:formed\:= 640.}$

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