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
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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)
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