Question

The areas of three adjacent faces of a cuboid are x, y, and z. If the volume is V, prove that V2 = xyz.​

Answer 1

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

Let assume that

↝ Length of cuboid be l units

↝ Breadth of Cuboid be b units

↝ Height of Cuboid be h units.

We know, Area of three adjacent faces are given by

↝ Area of first face = l × b

↝ Area of second face = b × h

↝ Area of third dace = h × l

According to statement,

The area of three adjacent faces be x, y, z respectively.

So,

$\rm \implies\:l \times b = x - - - - (1)$

$\rm \implies\:b \times h = y - - - - (2)$

$\rm \implies\:h \times l = z - - - - (3)$

On multiply equation (1), (2) and (3), we get

$\rm :\longmapsto\:xyz = {l}^{2} \times {b}^{2} \times {h}^{2}$

$\bf :\longmapsto\:xyz = {(lbh)}^{2} - - - - (4)$

We know,

Volume of cuboid having length l, breadth b and height h respectively, is given by

$\bf :\longmapsto\:V \: = \: lbh - - - - (5)$

So, on substituting equation (5) in (4), we get

$\rm :\longmapsto\:xyz = {V}^{2}$

Thus,

$\rm \implies\:\boxed{ \tt{ \: {V}^{2} = xyz \: }}$

Hence, Proved

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Additional Information

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)

Answer 2

Answer:

Let the 3 dimensions of the cuboid be l,b and h

So,

x=lb

y=bh

z=hl

multiplying above three equations,

xyz=lb×bh×hl=l2b2h2

As,

V=lbh

So,

V2=l2b2h2V2=xyz

Hence Proved

Step-by-step explanation:

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