Final answer:
To determine the minimum balloon diameter required, we need to consider the weight of the load and the buoyant force provided by the helium inside the balloon. We can use the ideal gas law and the equation for the volume of a sphere to calculate the minimum required diameter.
Step-by-step explanation:
To determine the minimum balloon diameter required, we need to consider the weight of the load and the buoyant force provided by the helium inside the balloon.
The buoyant force on the balloon can be calculated using Archimedes' principle:
Buoyant force = Weight of the displaced air
The weight of the displaced air is equal to the weight of the load, which is 10 metric tons (10,000 kg).
We can then use the density of air at 21°C and atmospheric pressure (1.01325 × 10^5 Pa) to calculate the volume of air displaced:
ρ = mass/volume
Using the ideal gas law, we can convert the density of air to the density of helium at the same conditions:
PV = nRT
Finally, we can use the equation for the volume of a sphere to find the minimum required diameter:
V = (4/3)πr^3
Putting it all together, we can substitute the values and solve for the minimum balloon diameter.