**Introduction:**
When an empty plastic box is placed deep inside water, it experiences an upward buoyant force that reduces its net weight. This buoyant force causes the box to accelerate upwards. To make the box accelerate downwards at the same rate, we need to add some mass to the box, such as sand, to increase its weight.

**Determining the Weight of the Box:**
Before adding sand to the box, we need to determine the weight of the box when it is empty. The weight of an object is given by the formula:

Weight = Mass * Acceleration due to gravity (g)

Since the box is accelerating upwards at a rate of g/6, its net weight is:

Weight = Mass * (g - g/6) = Mass * (5g/6)

**Calculating the Weight of the Sand:**
To make the box accelerate downwards at the same rate, we need to increase its weight by adding sand. Let's assume the mass of the sand added to the box is 'm_sand'. The weight of the sand can be calculated using the same formula:

Weight of Sand = Mass of Sand * Acceleration due to gravity (g)

Since we want the box to accelerate downwards at the rate of g/6, the net weight of the box and sand should be:

Weight of Box + Sand = (Mass + Mass of Sand) * (g - g/6) = (Mass + Mass of Sand) * (5g/6)

**Equating the Weights:**
To find the mass of the sand required, we can equate the net weight of the box and sand with its weight when empty:

(Mass + Mass of Sand) * (5g/6) = Mass * (5g/6)

**Simplifying the Equation:**
By canceling out the common terms and dividing both sides by (5g/6), we get:

Mass + Mass of Sand = Mass

Mass of Sand = 0

From this equation, we can conclude that no sand needs to be added to the box for it to accelerate downwards at the rate of g/6. The weight of the empty box is sufficient to achieve the desired acceleration.

**Conclusion:**
In this scenario, we found that no sand needs to be added to the empty plastic box for it to accelerate downwards at the rate of g/6. The weight of the empty box itself is enough to achieve the desired acceleration.