Question

Three metal cubes of sides 3cm , 4cm and 5 cm are melted to form a larger cube . Find volume, total surface area and edge length of new cube formed.

Answer 1

Answer:

The volume of the new cube formed is 216 cm³.

The edge length of the new cube is 6 cm.

The total surface area of the new cube is 216 cm².

Step-by-step-explanation:

We have given that,

Edge lengths of cubes are 3 cm, 4 cm and 5 cm.

The thee cubes are melted to form a larger cube.

We have to find the volume, total surface area and edge length of the new cube.

As the three cubes are melted to form a new cube, the sum of the volumes of the three cubes must be equal to the volume of resulting new cube.

∴ Volume of cube 1 + Volume of cube 2 + Volume of cube 3 = Volume of new cube

We know that,

Volume of cube = ( Length of edge )³

⇒ ( 3 )³ + ( 4 )³ + ( 5 )³ = Volume of new cube

We know that,

a³ + b³ + c³ = ( a + b + c ) ( a² + b² + c² - ab - bc - ac ) + 3abc

⇒ Volume of new cube = ( 3 + 4 + 5 ) [ 3² + 4² + 5² - ( 3 * 4 ) - ( 4 * 5 ) - ( 3 * 5 ) ] + ( 3 * 3 * 4 * 5 )

⇒ Volume of new cube = 12 ( 9 + 16 + 25 - 12 - 20 - 15 ) + ( 9 * 20 )

⇒ Volume of new cube = 12 ( 25 - 20 + 16 - 12 - 15 + 9 ) + 180

⇒ Volume of new cube = 12 ( 5 + 4 - 6 ) + 180

⇒ Volume of new cube = 60 + 48 - 72 + 180

⇒ Volume of new cube = 180 + 60 - 72 + 48

⇒ Volume of new cube = 240 - 24

⇒ Volume of new cube = 216 cm³

∴ The volume of the new cube formed is 216 cm³.

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Now,

Volume of new cube = ( Length of edge )³

⇒ 216 cm³ = ( Length of edge )³

Taking cube root on both sides, we get,

⇒ Length of edge = ∛216 cm

⇒ Length of edge = ∛( 4 * 54 )

⇒ Length of edge = ∛( 4 * 9 * 6 )

⇒ Length of edge = ∛( 2 * 2 * 3 * 3 * 6 )

⇒ Length of edge = ∛( 2 * 3 * 2 * 3 * 6 )

⇒ Length of edge = ∛( 6 * 6 * 6 )

⇒ Length of edge = 6 cm

∴ The edge length of the new cube is 6 cm.

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Now, we know that,

Total surface area of cube = 6 ( Edge length )²

⇒ Total surface area of cube = 6 ( 6 )²

⇒ Total surface area of cube = 6 * 36

⇒ Total surface area of cube = 216 cm²

∴ The total surface area of the new cube is 216 cm².