Answer:
length = 10.64 cm, width = 5.64 cm, and height = 5 cm.
Step-by-step 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[tex]w^{2}[/tex] + 25w
volume of the box = 5[tex]w^{2}[/tex] + 25w
Since the box was completely filled by 150 dice, then;
total volume of dice = volume of the box
300 = 5[tex]w^{2}[/tex] + 25w
5[tex]w^{2}[/tex] + 25w - 300 = 0
Divide through by 5 to have;
[tex]w^{2}[/tex] + 5w - 60 = 0
Applying the quadratic expression,
w = (-b ± [tex]\sqrt{b^{2} - 4ac}[/tex]) ÷ 2a
where: a = 1, b = 5 and c = -60
(-5 ± [tex]\sqrt{5^{2} - 4(1*-60)}[/tex] ) ÷ 2
(-5 ± [tex]\sqrt{265}[/tex]) ÷ 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.
