A lost shipping container is found resting on the ocean floor and completely submerged. The container is 6.0 m long, 2.2 m wide, and 2.3 m high. Salvage experts attach a spherical balloon to the top of the container and inflate it with air pumped down from the surface. When the balloon's radius is 1.7 m, the shipping container just begins to rise towards the surface. What is the mass of the container? Ignore the mass of the balloon and the air within it. Do not neglect the buoyant force exerted on the shipping container by the water. The density of seawater is 1025 kg/m3.

I am not sure I have ever seen a really watertight shipping container but anyway:)

Volume of container = 6*2.2*2.3 = 30.36 m^3
Buoyancy of container = 30.36*1025 g N = 31,119 g N

Buoyancy of lift bag = 1025 g (4/3) pi r^3 = 21,094 g N

total up force = 52,213 g N

total down force = mass of container * g

so
mass of container = 52,213 kg

I used another method of solving this problem. I also tried using your method and let me tell you that it really worked.

Oh buoy, we have a container conundrum! Let's dive into the problem, shall we?

To find the mass of the container, we need to figure out the buoyant force acting on it. The buoyant force is equal to the weight of the water displaced by the object.

The volume of the container is given by its dimensions (length, width, and height):

Volume = length x width x height
Volume = 6.0 m x 2.2 m x 2.3 m

Now, the volume of water displaced by the container is the same as the volume of the balloon:

Volume of balloon = (4/3)πr³
Volume of balloon = (4/3) x 3.14 x (1.7 m)³

So, the buoyant force can be calculated as:

Buoyant force = density of seawater x volume of balloon x acceleration due to gravity

Now, we know the density of seawater is 1025 kg/m³, and the acceleration due to gravity is approximately 9.8 m/s². Plugging those values along with the volumes we calculated earlier:

Buoyant force = 1025 kg/m³ x [(4/3) x 3.14 x (1.7 m)³] x 9.8 m/s²

The buoyant force is acting upwards, which is why the container starts to rise. Now, we can equate this force to the weight of the container:

Buoyant force = mass of the container x acceleration due to gravity

Rearranging the equation, we can solve for the mass of the container:

Mass of the container = Buoyant force / acceleration due to gravity

Calculating this gives you the mass of the container. But hey, wouldn't it be easier to just lift the container with some super-strong comedic muscles? *wink*

To find the mass of the container, we need to calculate the total buoyant force acting on it when the balloon is inflated. The buoyant force is equal to the weight of the water displaced by the submerged portion of the container.

First, let's calculate the volume of the submerged portion of the container. Since the container is rectangular, the submerged volume can be found by multiplying the length, width, and height of the container.

Volume of the submerged portion = Length x Width x Height
= 6.0 m x 2.2 m x 2.3 m

Next, we need to calculate the weight of the displaced water. The weight is equal to the volume of water displaced multiplied by the density of seawater (1025 kg/m^3) and the acceleration due to gravity (9.8 m/s^2).

Weight of the displaced water = Volume of the submerged portion x Density of seawater x Acceleration due to gravity
= (6.0 m x 2.2 m x 2.3 m) x (1025 kg/m^3) x (9.8 m/s^2)

The buoyant force acting on the container will be equal to the weight of the displaced water. Therefore, the mass of the container is equal to the weight of the displaced water divided by the acceleration due to gravity.

Mass of the container = Weight of the displaced water / Acceleration due to gravity
= [(6.0 m x 2.2 m x 2.3 m) x (1025 kg/m^3) x (9.8 m/s^2)] / (9.8 m/s^2)

Simplifying the equation, we can cancel out the acceleration due to gravity:

Mass of the container = (6.0 m x 2.2 m x 2.3 m x 1025 kg/m^3)

Calculating this expression will give us the mass of the container.