# A juice box measures 6.4 cm by 4.0 cm by 10.5 cm. It conains 250 ml of juice .The manufacturer wants to design a larger box by increasing each dimension by the same amount.Suppose the larger box...

A juice box measures 6.4 cm by 4.0 cm by 10.5 cm. It conains 250 ml of juice .

The manufacturer wants to design a larger box by increasing each dimension by the same amount.Suppose the larger box holds twice as much juice. What are it's dimentions

Please explain it with detail

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### 2 Answers

The dimensions for the small box = 6.4* 4 * 10.5

==> the volume of the box = 268.8

Let x be the ratio for the increase of the dimenstion.

Then, new dimensions are:

(6.4)x * 4x * (10.5)x = (268.8)x^3

However, we know that the volume of the new box should contain 500 ml

Then the volume for the larger container should be a little more that 500 cm^3 ( 1 cm^3 = 1 ml)

==> (268.8)x^3 = 500

==> x= 1.23 (approx.)

Then new dimensions are :

**==> 7.872 * 4.92 * 12.915 = 500.2 cm^3 **

The dimension of the box = 6.4cm by 4cm by 10.5cm

Therefore the volume of the box = 250 ml is proportional to 6.4*4*10.5cm.

So 250 = k(6.4)(4)(10.5).....(1), where k is a constant of proportion.

The box with twice the volume has the volume 500 ml and since the shape is similar, we presume the factor of increase in its dimension is x. So box should measure 6.4x by 4x by 10.5.

Therefore 500 = k(6.4x)(4x)(10.5x)...(2)

(1)/(2) gives:

250/500 = k(6.4)(4)(10.5)/k(6.4)(4)(10.5)x^3.

1/2 = 1/x^3.

Multiply by 2x^3 :

x^3 = 2.

x = 2(1/3).

Therefore the measure dimensions of the larger box = 6.4(6.4*2^1/3) by (4*2^1/3) by (10.5*2^1/3)

= 8.0635cm by 5.0397cm by 13.2292cm