a 250 ml container of fruit drink measures 6.5cm by 4cm by 10cm. the tcarton manufacturer would like to do design a 1L box with each dimension...

If the volume is increased by the same amount, then the
designed volume is 250ml+250ml = 500ml. Then the tarcton has to increase the dimensions
by a factor x. So the container dimension is 6*6x, 4x and
10x.

Since both shapes are similar,the volumes 250 ml and
500 ml  are proportional to the measures 6.5 by 4 by 6.5x by 4x by
10x

 250 ml =k*6.5*4*10..........(1)
and

500ml = k(6.5x)(4x)(10x).........(2)., where k is the
constant of proportionality.

We eliminate k  by dividing eq
(1) by eq (2):

250/500 =
k6.5*4*10/(k*6.5x*4x*10x)

1/2 =
1/x^3

Nultiply by 2x^3.

x^3 =
2

x = 2^(1/3) =
1.25992105

Therefore the  dimensions to be designed by the
manufacturer =

(6.5x , 4x , 10x) = 8.195 , 5.0397 ,
12.5992)