Question

(iii) The dimensions of a metallic cuboid are 44 cm x 42 cm x 21 cm. It is melted and
recast into a sphere. Find the surface area of the sphere.​

Answer 1

Answer:

The surface area of the sphere is 5544 cm².

Step-by-step explanation:

Solution :

Dimensions = 44 cm × 42 cm × 21 cm

Melted into = Sphere

Here,

Volume of the cuboid = Volume of the sphere

$\bigstar \: \boxed{\sf{Volume \: of \: cuboid = l \times b \times h}}$

• l = 44 cm
• b = 42 cm
• h = 21 cm

$\sf{\longrightarrow} \: 44 \times 42 \times 21$

$\sf{\longrightarrow} \: 1848 \times 21$

$\sf{\longrightarrow} \: 38808 \: {cm}^{3}$

Volume of the cuboid = 38808 cm³.

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$\bigstar \: \boxed{\sf{Volume \: of \: sphere= \frac{4}{3}\pi{r}^{3}}}$

$\sf{\longrightarrow} \: \dfrac{4}{3} \times \dfrac{22}{7} \times {(r)}^{3} = 38808$

$\sf{\longrightarrow} \: \dfrac{88}{21}\times {(r)}^{3} = 38808$

$\sf{\longrightarrow} \: 88r^{3} = 38808 \times 21$

$\sf{\longrightarrow} \: 88r^{3} = 814968$

$\sf{\longrightarrow} \: r^{3} = {\dfrac{814968}{88}}$

$\sf{\longrightarrow} \: r^{3} = 9261$

$\sf{\longrightarrow} \: r = \sqrt[3]{9261}$

$\sf{\longrightarrow} \: r = 21$

Radius of the sphere = 21 cm

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$\bigstar \: \boxed{\sf{Surface \: area \: of \: sphere =4\pi{r}^{2}}}$

$\sf{\longrightarrow} \: 4 \times \dfrac{22}{7} \times {(21)}^{2}$

$\sf{\longrightarrow} \: 4 \times \dfrac{22}{7} \times 21 \times 21$

$\sf{\longrightarrow} \: 4 \times 22\times 21 \times 3$

$\sf{\longrightarrow} \:88\times63$

$\sf{\longrightarrow} \: 5544 \: {cm}^{2}$

Therefore, the surface area of the sphere is 5544 cm².

Answer 2

$\star~\boxed{ \sf{ Volume~ of~ cuboid = Volume~ of ~sphere}}$

$\circ~ \sf lbh = \dfrac{ 4}{3 } πr³$

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$~~ \sf \dfrac{ 4}{3 }πr³ = 44×42×21$

$~~ \sf r³=44 \times 42 \times 21 \times \dfrac{3}{4} \times \dfrac{7}{22}$

$~~ \sf r³=21 \times 21 \times 21$

$~~ \sf r= \sqrt[3]{21 \times 21 \times 21}$

$~~ \star ~\sf \underline{\underline{ r = 21 \: cm}}$

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$\circ~ \sf Surface \: area \: of \: sphere = 4πr²$

$\sf SA=4 \times \dfrac{22}{7} \times (21)²$

$\sf SA=4 \times 22 \times 3×21$

$\star~ \sf \underline {\underline{SA=5544 \: {cm}^{2}}}$

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