Question

three cubes of a metal whose edges are in the ratio 3:4:5 are melted and converted into a single cube whose diagonal is 12√3 cm. find the volumes of all the cubes.

Answer 1

Let the edges of the three cubes = 3x, 4x and 5x
Then,
Their volume =(3x)^3,(4x)^3,(5x)^3
The length of the diagonal of the single cube =12√3
As we know that,the length of the diagonal of a cube of side a=a√3
⇒ edge of the single cube = 12 cm
Now, Volume of the single cube = (12 cm)3 = 1728, cm3
Since the three cubes are melted and converted into single cube
⇒ Sum of volumes of three cubes = volume of single cube
27x^3+64x^3+125x^3=1728
216x^3=1728
x^3=8
x=2
∴ The edges of the three cubes are 6cm, 8cmand 10 cm
now find volume of cube by putting value of x
27(6)^3,64(8)^3,125(10)^3
I hope it helps you

Answer 2

let common ratio of sides be x.
thus, sides of small cubes are 3x, 4x and 5x.
total volume of 3 metal cubes = 27x³ +64x³+125x³=216x³
let side of big cube=a
we know: diagonal of cube=a√3
thus side of big cube=12cm
now volume of big cube = volume of 3 small metal cubes
12³=216x³
1728=216x³
x³=1728÷216
x³=8
x=2cm
so volume of 1st metal cube=(3×2)³=216cm³
volume of 2nd metal cube=(4×2)³=512cm³
volume of 3rd metal cube=(5×2)³=1000cm³
volume of big cube=12³=1728cm³