[leot]

(According to GPT4)

Using the hydrostatic pressure formula:

Pressure = Density of water × Acceleration due to gravity × Depth

Assuming the same density of water (1,000 kg/m³) and acceleration due to gravity (9.8 m/s²):

Pressure = 1,000 kg/m³ × 9.8 m/s² × 4,000 m Pressure = 39,200,000 Pascal (Pa)

Converting to atmospheres:

Pressure in atmospheres = 39,200,000 Pa / 101,325 Pa/atm Pressure in atmospheres ≈ 387 atm

Considering a safety factor of 2, as before, we can calculate the required thickness:

Required thickness = Pressure / (Tensile strength × Safety factor) Required thickness = 387 atm × 101,325 Pa/atm / (434 MPa × 2)

Converting units:

Required thickness ≈ 39,200,000 Pa / (434 × 10^6 Pa × 2) Required thickness ≈ 0.045 meters or 1.48 inches

Therefore, at a depth of 4,000 meters, the titanium walls of the submersible sphere with a 10 ft diameter would need to be approximately 1.48 inches thick to resist implosion, assuming a safety factor of 2. Again, please note that this estimation may vary based on the design, shape, and structural considerations of the submersible sphere. Consulting with experts is essential for accurate calculations and safety assessments.

As for amount of titanium required:

First, let's convert the wall thickness to meters. Since 1 inch is approximately 0.0254 meters, the wall thickness of 1.5 inches would be approximately 0.0381 meters.

Now, let's calculate the internal volume of the sphere by subtracting the volume of the inner sphere (10 ft diameter minus 2 times the wall thickness) from the volume of the outer sphere (10 ft diameter):

Inner radius = Radius of the sphere - Wall thickness Inner radius = 1.524 meters - 0.0381 meters Inner radius = 1.4869 meters

Inner volume = (4/3) * π * (Inner radius)^3 Inner volume ≈ 14.012 cubic meters

Outer volume = (4/3) * π * (1.524 meters)^3 Outer volume ≈ 14.137 cubic meters

Now, we can calculate the volume of the titanium walls by subtracting the inner volume from the outer volume:

Titanium wall volume = Outer volume - Inner volume Titanium wall volume ≈ 14.137 cubic meters - 14.012 cubic meters Titanium wall volume ≈ 0.125 cubic meters

Finally, we can calculate the mass of titanium using the density of titanium (4,506 kg/m³):

Mass of titanium = Titanium wall volume * Density of titanium Mass of titanium ≈ 0.125 cubic meters * 4,506 kg/m³ Mass of titanium ≈ 563.25 kg

Therefore, with a wall thickness of 1.5 inches, the approximate amount of titanium required for the walls of the submersible sphere with a 10 ft diameter would be approximately 563.25 kilograms.

Titanium is currently around $6/kg, so ~$3.4k for just the titanium that made up the wall.