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Boyle's law is an essential principle in physics that describes the inverse relationship between the pressure and the volume of a gas, at a constant temperature. In simpler terms, if you increase the pressure on a gas, its volume decreases, and conversely, if you decrease the pressure, its volume increases.

In the context of the exercise, the tank containing helium gas at 150 atm is at a much higher pressure compared to the surrounding air pressure. When we want to inflate balloons to a lesser pressure of 1.20 atm, we can apply Boyle's law to find out how much space the gas will take up at this new pressure. The formula used here translates the initial volume (\(V_1\) measured under high pressure (\(P_1\) to the new volume (\(V_2\) under lower pressure (\(P_2\) which is necessary to calculate the volume of gas we can use to inflate the balloons.

It's exact applications like this that underscore Boyle's law as foundational for understanding how gases behave under different pressures — from filling balloons to more complex tasks like operating pneumatic systems.

Calculating the volume of helium gas available for filling balloons involves applying the formula provided by Boyle's law, \(V_2 = V_1 * (P_1 / P_2)\). The first step is identifying the known variables from the problem: the initial volume (\(V_1\) of 0.100 m³ and the initial pressure (\(P_1\) at 150 atm. Next, we determine our desired final pressure (\(P_2\) to be 1.20 atm. Upon substituting these numbers into Boyle's law, we calculate the new volume (\(V_2\).This calculation results in a much larger volume of 12.5 m³, demonstrating how lower pressure allows the helium gas to expand. It is important to grasp how significantly gas can spread out when transitioning from high to low pressure, an insight which also applies to various real-world scenarios like weather balloons' ascension through the atmosphere where the air pressure decreases with altitude.

For a discrete object like a balloon shaped as a sphere, the volume can be determined using the formula for the volume of a sphere: \(V = (4/3)Ï€r^3\). To apply this formula, one must first obtain the sphere's radius, which is half the diameter. In the given exercise, each balloon has a diameter of 0.300 m, thus the radius is 0.150 m.

After substituting the radius into the volume formula, we obtain the volume of a single balloon. With this result, we can then figure out how many such balloons can be filled using the total available volume of helium calculated previously. It's crucial to execute each step precisely, including squaring the radius — it's a common mistake to overlook the cube in the formula. Understanding the geometry of objects is key in solving real-world problems where volumes need to be calculated, such as estimating the capacity of tanks or measuring the amount of liquid needed to fill a container.