Question

A rectangular is completety filled with dice.Each dice has volume of 2cm³.The length of the box is 5cm grater than its width and its height is 5cm.Suppose the box holds at most 150dice.What are the posibble dimensions of the box​

Answer 1

length = 10.64 cm, width = 5.64 cm, and height = 5 cm.

Explanation:

Total volume of dice that filled the box = (150 x 2) cm³

= 300 cm³

Let the length, width and height of the box be represented by l, w and h respectively.

So that;

l = (w + 5) cm

w = w

h = 5 cm

volume of the box = length x width x height

= (w + 5) x w x 5

= (w + 5) x 5w

= 5
`w^(2)` + 25w

volume of the box = 5
`w^(2)` + 25w

Since the box was completely filled by 150 dice, then;

total volume of dice = volume of the box

300 = 5
`w^(2)` + 25w

5
`w^(2)` + 25w - 300 = 0

Divide through by 5 to have;

`w^(2)` + 5w - 60 = 0

Applying the quadratic expression,

w = (-b ±
`\sqrt{b^(2) - 4ac}`) ÷ 2a

where: a = 1, b = 5 and c = -60

(-5 ±
`\sqrt{5^(2) - 4(1*-60)}` ) ÷ 2

(-5 ±
`√(265)`) ÷ 2

(-5 ± 16.28) ÷ 2

(-5 + 16.28) ÷ 2 OR (-5 - 16.28) ÷ 2

5.64 OR -10.64

Thus, w = 5.64 cm

So that, l = (w + 5) = 10.64 cm

The possible dimensions of the box are: length = 10.64 cm, width = 5.64 cm and height = 5 cm.