This is a fairly straight-forward question, providing us with a force vector that is acting upon a surface and asking us for the pressure.

First and foremost, we can examine the equation for pressure:

Next, we'll need to figure out the sign convention for pressure. Based on the above equation, it may seem like if we plug in a negative value for pressure and a positive value for area, we'll get a negative value for pressure. However, this is not the case.

As it turns out, pressure is a scalar quantity. The reason for this is that when any force acts on a surface, it is acting perpendicularly to that surface. Therefore, the direction in which the force is acting is dependent on how the surface is oriented. No matter which way you orient the surface, the force that is acting on that surface will always be perpendicular. Consequently, pressure is essentially just a proportionality between force and area that has no bearing on direction.

To answer this question, it's useful to think of this conceptually as the balloon being "submerged" in the atmosphere. Remember, buoyant forces refer to the force caused by the displacement of a fluid, and both liquids and gasses count as fluids. Thus, in this case, the balloon can be thought of as submerged in a fluid in the same way that any object can be submerged in a liquid such as water.

We're trying to look for the point at which the balloon stops rising. This is analogous to a situation in which the balloon is "floating" in the fluid. In such a case, the upward and downward forces are equal and balance each other out.

So now, we need to identify the upward and downward forces acting on the balloon and set them equal to each other. The upward force will be the buoyant force due to the displacement of air from the atmosphere. The downward force will be the weight of the balloon.

We can further rewrite the mass of the balloon in terms of its weight and density.

Canceling out the gravity term, we obtain:

Next, it's important to realize that the volume of atmosphere being displaced is exactly the same as the volume of the balloon, since it is the balloon that is causing the displacement of air in the atmosphere. Thus:

Furthermore, because these two values are equal, we can cancel them out in the above expression to obtain the following.

Here, we have shown that once the density of the balloon becomes equal to that of the surrounding atmosphere, it will cease rising.

In this question, we're told that three containers of different shapes are filled with water. We're also told that each container has the same depth of water.

To find the pressure at the bottom of any of the containers, we'll need to remember the equation for pressure.

Since each of the containers is open to the atmosphere, we can disregard the term for atmospheric pressure in the above expression. Therefore, the absolute pressure becomes the gauge pressure.

Also, since each container is filled with water, the density of the fluid in each container is identical. Moreover, the depth we are considering for each container is also the same. Therefore, the pressure at the bottom of each container is exactly the same.

The total pressure on the ball includes both hydrostatic and atmospheric pressure:

We are given the atmospheric pressure, so we just need to determine the hydrostatic pressure using the following expression:

Plugging in values:

Therefore,

Now to determine the force on the ball, we need it's surface area. For a sphere:

Then,

We can calculate the total pressure on the ball from the given force and radius:

Where for a sphere:

There are two pressures that combine to the total pressure on the ball: hydrostatic and atmospheric. 

Since the ball is submerged in the ocean, we know that the surface of the water is at sea level and thus has a pressure:

Also, we can calculate the hydrostatic pressure with the following expression:

Where density is of the ocean and the height is the depth of the ball. Plugging this into our last expression, we get:

Rearranging for height:

To determine the hydrostatic pressure, we need to know the depth of the ship at time t = 12s. We can determine this with the following expression:

Then using the expression for hydrostatic pressure:

Given the hydrostatic pressure, we can calculate the depth that this occurs at:

Rearranging for height:

Plugging in our values, we get:

We can then use the following kinematics equation to determine how much time has passed:

If we designate the downward direction as positive and plugging in values to the kinematics equation, we get:

Rearranging, we get:

Since we can't have a negative time, the first one is the answer.

We will use the expression for hydrostatic pressure for this problem:

In this scenario, the pressures from each material are additive. Therefore:

Plugging in expressions:

Plugging in values:

Note how we used 0.9m for the height of mercury since we are asked for the pressure at a height of 0.5m.

The following formula on pressure and area is used:

We substitute our known values and solve for F2 to obtain the output force:

Therefore the correct answer is  of force.

The formula for pressure is given as:

Where  is pressure in pascals,  is force in newtons, and  is area in meters squared. By substituting our known values we can solve for pressure:

Therefore the correct answer is