The smallest possible length of the straw that cannot accidentally get stuck in the juice box is 6 cm.
To ensure the straw doesn't accidentally get stuck in the juice box, we can use the concept of the Pythagorean theorem to find the shortest possible length of the straw that fits within the box.
Let's assume the dimensions of the juice box are as follows:
Length = 15 cm
Width = 3 cm
Height = 5 cm
The shortest length of the straw that fits within the box without getting stuck can be found by calculating the diagonal of the box. Using the Pythagorean theorem (a^2 + b^2 = c^2, where c is the hypotenuse):
a = Width = 3 cm
b = Height = 5 cm
Diagonal (c) = √(Width^2 + Height^2)
Diagonal (c) = √(3^2 + 5^2)
Diagonal (c) = √(9 + 25)
Diagonal (c) = √34
Diagonal (c) ≈ 5.83 cm (approximately)
Since the length of the straw must be an integer value, the smallest possible integer greater than 5.83 cm that still ensures it won't get stuck in the box is 6 cm. Therefore, the smallest possible length of the straw that cannot accidentally get stuck in the juice box is 6 cm.