The atmospheric re-entry temperature on the surface of a space capsule represented by the ellipsoid 4x^2 + y^2 + 4z^2 = 16 is given by T = 8x^2 + 4yz −16z + 600 (°C ) .
The material from which the capsule is made can only withstand a temperature of 650°C. Will the occupants of the space-craft survive re-entry?
The temperature is: T(x,y,z) = 8x^2 + 4yz – 16z + 600
The constraint is: f(x,y,z) = 4x^2 + y^2 + 4z^2 – 16 = 0
We wish to maximise the temperature T() subject to the constraint f().
The Langrangian is: L(x,y,z,λ) = T(x,y,z) – λ.f(x,y,z)
Taking the partial differentials,
Tx = λ.fx; 16x = λ.8x à λ = 2
Ty = λ.fy; 4x = λ.2y à 2z = λy à 2z = 2y à y = z
Tz = λ.fz; 4y – 16 = λ.8z à y – 4 = 2λz à y – 4 = 4y à y = -4/3, z = -4/3
f = 0; 4x^2 + y^2 + 4z^2 – 16 = 0
4x^2 + 5(-4/3)^2 = 16
4x^2 + 80/9 = 16
x^2 + 20/9 = 4
x^2 = (36 – 20)/9 = 16/9
x = ± 4/3
Max temp occurs at the positions (x,y,z) = (-4/3, -4/3, -4/3), (4/3, -4/3, -4/3)
Value of this max temp is: 8x^2 + 4yz – 16z + 600 = 8(4/3)^2 + 4(4/3)^2 – 16(4/3) + 600
Tmax = 12(16/9) – 64/3 + 600 = 600 ⁰C
Since Tmax is less than 650 ⁰C, then the astronauts will survive re-entry