Question

Three cubes of metal whose edges are in the ratio 3 : 4 : 5 are melted down into a single cube whose diagonal is 12√3 cm. Find the edges of three cubes.

Answer 1

Answer:

The edge of first cube 6 unit

The edge of second cube 8 unit

The edge of third cube 10 unit

Step-by-step explanation:

Given as :

The ratio of edge of three small cubes = 3 : 4 : 5

Let The edge of first cube = 3 x

Or, volume of first cube = $v_1$ = (edge)³

i.e   $v_1$ = (3 x)³

Or,  $v_1$ =  27 x³

The edge of second cube = 4 x

Or, volume of second cube = $v_2$ = (edge)³

i.e   $v_2$ = (4 x)³

Or,  $v_2$ =  64 x³

The edge of third cube = 5 x

Or, volume of second cube = $v_3$ = (edge)³

i.e   $v_3$ = (5 x)³

Or,  $v_3$ =  125 x³

Again

Three small cubes are melted to recast a bigger cube

Let, volume of bigger cube = V

A/Q diagonal of bigger cube = D = √3 a

where a is the edge

Or, 12 √3  = √3 a

∴   a = $\dfrac{12\sqrt{3} }{\sqrt{3} }$

i.e edge of bigger cube = a = 12 unit

So, volume of bigger cube = (edge)³

i.e  v = 12³

∴   v =  1728 cubic unit

Now,

Volume of bigger cube = sum of all three small cubes

i.e  v = $v_1 + v_2 + v_3$

Or,  12³  =  (3 x)³  +  (4 x)³  + (5 x)³

Or,  12³ = 27 x³ + 64 x³ + 125 x³

or,  12³ = 216 x³

Or,  12³ =  (6 x)³

removing cube both side

∴   x = $\dfrac{12}{6}$

i.e x = 2

So, The edge of first cube = 3 x = 3 × 2 = 6 unit

     The edge of second cube = 4 x = 4 × 2 = 8 unit

     The edge of third cube = 5 x = 5 × 2 = 10 unit

Hence,

The edge of first cube 6 unit

The edge of second cube 8 unit

The edge of third cube 10 unit    Answer

Answer 2

answer : 6cm , 8cm and 10 cm

Three cubes of metal whose edges are in the ratio 3 : 4 : 5 are melted down into a single cube whose diagonal is 12√3 cm.

let proportionality constant is x

then, edge length of three cubes are 3x , 4x and 5x respectively.

so, sum of volume of three cubes = volume of bigger cube

⇒(3x)³ + (4x)³ + (5x)³ = (12√3/√3)³ [ edge length = diagonal/√3 ]

⇒27x³ + 64x³ + 125x³ = 12 × 12 × 12

⇒216x³ = 12 × 12 × 12

⇒18x³ = 12 × 12

⇒x³ = 144/18 = 8

⇒x = 2

hence edge length of cubes ; 3 × 2 , 4 × 2 , 5 × 2

i.e., 6 cm, 8cm and 10 cm