Question

A helium balloon ride lifts up passengers in a basket. Assume the density of air is 1.28 kg/m³ and the density of helium in the balloon is 0.18 kg/m³. The radius of the balloon (when filled) is R = 4.5 m. The total mass of the empty balloon and basket is m_b = 123 kg and the total volume is V_b = 0.068 m³. Assume the average person that gets into the balloon has a mass m_p = 74 kg and volume V_p = 0.077 m³.

Answer 1

Problem One: volume
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V = 4/3 pi r^3
V = 1.3333 * 3.141592 * 4.9 ^ 3
V = 492.8 m^3

Problem Two: Force
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Mass balloon = 125 kg
Mass Helium = d_Helium * Volume = 0.18 kg/m^3 * 492.8 = 88.71 kg
Total Mass = 213.71 kg

Force = m*g
m = 213.71
g = 9.81
F = m*g
F = 213.71 * 9.81
F = 2096.5 N

Problem Three: Buoyant Force
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Mass of displaced volume of air. = d * V
d = 1.28 kg/m^3
V = 492.8 m^3
Mass = 1.26 * 492.8
Mass = 621 kg

The volume of the basket = 0.059 m^3
The buoyant mass = the volume of air displaced * density
The buoyant mass = 0.059m^3 * 1.28 kg/m^3 = 0.07552 kg

The buoyant mass is the difference between the balloon full of helium that displaces the air and the helium mass and the basket mass.
m = 621 - 213.7+ 0.07552 = 407.37

The force = 407.37 * 9.81 = 3996.4 N

Problem Four
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This is really tricky. I think you are intended to use the volume, but you really should not have to.

F person (no buoyancy) = 72 * 9.81 = 706.32 N
Buoyancy on each person = 0.9543

With buoyancy = 706.32 - 0.076 * 1.28 * 9.81 =705.4 N
the actual force is close enough without the buoyancy, but I gave you both.

The number of people who could be lifted by the system = 3996.4/705.4 = 5.66 people. In this case you have to round downward, so you could lift 5 people.