Modern commercial airliners are largely made of aluminum, a light and strong metal. But the fact that aluminum is cheap enough that airplanes can be made out of it is a bit of historical luck. Before the discovery of the Hall-Héroult process in , aluminum was as rare and expensive as gold. What would happen if airplanes had to be made of steel? The fuselage of the Boeing 747, which can carry passengers, is approximately a hollow aluminum cylinder without ends, long, wide, and thick (see sketch at right). The fuselage of an airplane Suppose this fuselage was made of steel (density ) instead of aluminum (density ), and let's say the average passenger has a mass of . We'll also assume the engines can't lift any greater mass than they already do. Calculate the number of passengers that the Boeing 747 could carry if its fuselage was made of steel.

 n = 291 people

Explanation:

To solve this interesting problem we are going to take some data from the tables, the density of 7075 aluminum is 2.8 103 kg / m³, the density of 316 stainless steel is 7980 kg / m³, where we have selected one of the most used alloys in for its resistance.

The average mass of a passenger is mo = 80kg and the Boeing 747 aircraft can carry 550 passengers and a crew of 18 people with a length of 70.66 m,

Let's start by calculating the mass of the empty plane and the mass of the passengers

         ρ_Al = m_Al / V

         m_Al = ρ_Al V

         m_passenger= #_pasajeros 80

         m_passenger = 550 m₀

The total mass of the plane is

          M = m_passengers + m_Al

If the plane is built face to face, the mass of this material is

        ρ_Acero = m_steel / V

the total mass of avino is

         M = m_pasajeros` + m_steel

As they indicate the total mass of the plane must remain constant

         m_passengers + m_Al = m_passengers` + m_steel

let's substitute the values

         550 m₀ + 2.8 103 V = n m₀ + 7.98 103 V

where V is the volume of the material used which is the same in both cases

         n = 550 mo + (2.8 - 7.98) 103 V = n mo

         n = n = 550 + (2.8 - 7.98) 103 V / mo

The calculation of the volume of material used depends on the thickness of the sheet and the resistance of the material, if it assumes that it has the same thickness, e = 1 cm, approximate the volume to the length of the circle by the length of the cylinder by the thickness

         V = (2π r) L e

  the diameter of the plane approximately the height of the person

         d = 1.8 m

         V = 2π (1.8 / 2) 70.66 1 10⁻²

         V = 3.9957 m³

  we substitute

          n = 550 - 5.18 10³ (3.9957) / 80

          n = 550 - 258.7

          n = 291 people