Question

if the areas of three adjacent faces of a rectangular block are in the ratio of 2:3:4 and its volume is 9000cm^3.then the length of the shortest side is
a)10cm b)15cm c)25cm d)20cm e)30cm

Answer 1

let the edges (dimensions) of block be a, b, and c.
     a*b : b*c : c*a = 2 : 3 : 4
 
       a * b = 2 k        b * c = 3 k          c * a = 4 k    for some k > 0

multiply all these :
           a * b * b * c * c * a  = a² b² c² = 24 k³

             volume of block = a b c =  9, 000  cm³

            Hence a² b² c² = 9, 000 * 9, 000  = 24 k³
 
                         k³ = 27,000,000/8
                 k = 300 / 2  = 150
 
         Hence a * b = 300          b * c = 450        c * a = 600
 
                 a = abc / b c = 9000 / 450 = 20 cm
               b =  abc / ca  =  9000 / 600  =  15 cm
             c =  abc / ab =  30 cm

so 15 cm is answer

Answer 2

Answer:

Let the edge of the cuboid be acm,bcm and ccm.

And, a<b<c

The areas of the three adjacent faces are in ratio 2:3:4

So,

ab:ca:bc=2:3:4 and its volume is 9000cm

3

We have to find the shortest edge of the cuboid

Since,

bc

ab

=

4

2

c

a

=

2

1

∴ c=2a

Similarly,

bc

ca

=

4

3

b

a

=

4

3

∴ b=

3

4a

Volume of cuboid,

V=abc

⇒ 9000=a(

3

4a

)(2a)

⇒ 27000=8a

3

⇒ a

3

=

8

27×1000

⇒ a=

2

3×10

∴ a=15cm

Now, b=

3

4a

=

3

4×15

=20

c=2a=2×15=30cm

∴ The length of the shortest edge is 15cm