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 of
increased by the same amount. What are the dimensions of this larger container?
please write how did you solve it
The volume for the small container (V1) = 250 ml
Now we need to exapnd the volume for the larger comtainer by 4 times==> V2 = 1 L = 1000 ml
Then dimensions for the small comtainer are:
6.5 *4 *10 = 260 cm^3 = 260 ml ( 1 cm^3 = 1 ml)
For the larger container we need to expand it by the same ratio.
==> the dimensions for the larger container are:
(6.5)x *(4)x * 10x = 260*x^3 cm^3
Now the volume of the larger cartoon we need should be equal or a little larger than 1 litre (1000 ML)
==> 260x^3 = 1000
==> x^3 = 1000/260
==> x= 1.57 (approx.)
So x= 1.57 is the minimum amount that we will expand the cartoon's dimension so it will contain 1000ml (1 L).
Then the new dimensions are:
(6.5 * 1.57 ) *( 4*1.57) * (10* 1.57) = 1006.17 cm^3
= 10.205 * 6.28 * 15.7 = 1006.17 cm^3
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)