Question

Three cubes of metal whose edges are in the ratio 3 : 4 : 5 are melted into a single cube, the length of whose longest diagonal is 48ď–3 m. Calculate the length of the edges of the three cubes.

Answer 1

First of all the diagonal given is not clear. As per my knowledge it should be

$48\sqrt{3}$

In which case, here is the solution

Let the sides of three cubes be 3x, 4x and 5x respectively (ratio is 3:4:5)

and side of new cube formed be S

these cubes are melted and formed a single cube, in which case volume of these three cubes should be equal to volume of single new cube formed

∴ vol of cube 1 + vol of cube 2 + vol of cube 3 = volume of new cube formed

(3x)³ + (4x)³ + (5x)³ = S³ (Volume of cube = side³)

27x³ + 64x³ + 125x³ = S³

S³ = 216x³

S = ∛(216x³)

S = 6x

Now, diagonal of new cube given is 48√3 m

And we know that length of diagonal of a cube = $\sqrt{3 side^2 }$

$\sqrt{3 (6x)^2} =48\sqrt 3$

$6x\sqrt 3 =48\sqrt 3$

6x = 48

x = 8

∴ sides of given cubes is

3x = 24 m

4x = 32 m

5x = 40 m