Question

A solid metal cuboid is melted and recast into a single cube. The ratio of the length, breadth and height of the
cuboid was 4:2:1 and its total surface area was 448 cm? What is the total surface area (in cm) of the new
cube formed?​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

The ratio of the length, breadth and height of the

• cuboid was 4:2:1.

and

• Total Surface Area of Cuboid = 448 cm^2.

So,

Let assume that

• Length of Cuboid, l = 4x cm

• Breadth of Cuboid, b = 2x cm

• Height of Cuboid, h = x cm

We know,

$\rm :\longmapsto\:\red{ \bf \: TSA_{(Cuboid)} =( l \times b + b\times h + h \times l)}$

$\rm :\longmapsto\:448 = 2(4x \times 2x + 2x \times x + x \times 4x)$

$\rm :\longmapsto\:224 = 8 {x}^{2} + {2x}^{2} + {4x}^{2}$

$\rm :\longmapsto\:224 = 14 {x}^{2}$

$\rm :\longmapsto\: {x}^{2} = 16$

$\bf\implies \:x = 4$

Hence,

Dimensions of Cuboid are

• Length of Cuboid, l = 4x = 16 cm

• Breadth of Cuboid, b = 2x = 8 cm

• Height of Cuboid, h = x = 4 cm

Now,

Since,

• This cuboid is melted and recast in to a single cube.

So,

• Volume of cuboid = Volume of cube

Let edge of the cube = a cm

We know,

$\rm :\longmapsto\:Volume_{(cuboid)} = l \times b \times h$

and

$\rm :\longmapsto\:Volume_{(cube)} = {a}^{3}$

As,

$\rm :\longmapsto\:Volume_{(cuboid)} = Volume_{(cube)}$

$\rm :\longmapsto\:8 \times 4 \times 2 = {a}^{3}$

$\rm :\longmapsto\:4 \times 4 \times 4 \times 2 \times 2 \times 2 = {a}^{3}$

$\rm :\longmapsto\: {a}^{3} = {4}^{3} \times {2}^{3}$

$\rm :\longmapsto\:a = 4 \times 2$

$\bf\implies \:a = 8 \: cm$

Now,

We know,

$\bf :\longmapsto\:TSA_{(Cube)} = {6a}^{2}$

$\rm \: \: = \: \: 6 \times 8 \times 8$

$\rm \: \: = \: \: 384 \: {cm}^{2}$

$\bf :\longmapsto\:TSA_{(Cube)} = 384 \: {cm}^{2}$

Additional Information

Cube: 

• A cube has six faces, eight vertices and twelve edges. All the faces of the cube are in square shape and have equal dimensions.

Cuboid: 

• A cuboid has six faces, eight vertices and twelve edges. The faces of the cuboid are parallel.

Formula's of Cube :-

• Total Surface Area = 6(side)²

• Curved Surface Area = 4(side)²

• Volume of Cube = (side)³

• Diagonal of a cube = √3(side)

• Perimeter of cube = 12 x side

Formula's of Cuboid

• Total Surface area = 2 (Length x Breadth + breadth x height + Length x height)

• Curved Surface area = 2 height(length + breadth)

• Volume of the cuboid = (length × breadth × height)

• Diagonal of the cuboid =√(l² + b² + h²)

• Perimeter of cuboid = 4 (length + breadth + height)