Optional: Fluids

Density ρ is the mass per unit volume of a material, and pressure P is the force acting perpendicular to a surface per unit area of that surface: a scalar, measured in pascals, with no direction of its own. Use the supplied topic model Y_out = rho*g*t/100000*cosd(theta), where Y_out is PRESSURE P = ρgt cos θ (kPa). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees; g = 9.81 m s⁻². theta is Ramp tilt θ, measured in °. Keep rho (Slab density ρ) = 2400 kg m⁻³; t (Slab thickness t) = 20 cm fixed. For this exercise use only 4.78125 ° ≤ theta ≤ 40.2188 ° on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 4.2177961 kPa, with a bounded reading uncertainty ±0.071193 kPa. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown theta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00354375 °, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Hydrostatic pressure is the pressure within a fluid at rest; it rises with depth at the rate ρg, because each layer must hold up the weight of the fluid above it. Use the supplied topic model Y_out = g*(rho*min(h,t)/1000 + max(h-t,0)), where Y_out is GAUGE AT DEPTH h (kPa). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; min(a,b) selects the smaller value; max(a,b) selects the larger value; g = 9.81 m s⁻². h is Probe depth h, measured in m. Keep t (Oil layer thickness t) = 1.5 m; rho (Oil density) = 800 kg m⁻³ fixed. For this exercise use only 0.425 m ≤ h ≤ 3.575 m on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 20.076165 kPa, with a bounded reading uncertainty ±1.8077 kPa. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown h by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315 m, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Atmospheric pressure is the force per unit area exerted by the weight of the air above a surface; barometers measure it directly, and manometers and most gauges measure other pressures relative to it. Use the supplied topic model Y_out = 101.325*exp(-34.22*z/tk), where Y_out is ISOTHERMAL p (kPa). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. z is Altitude z, measured in km. Keep tk (Air temperature T) = 288 K fixed. For this exercise use only 1.275 km ≤ z ≤ 10.725 km on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 43.899533 kPa, with a bounded reading uncertainty ±2.3351 kPa. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown z by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000945 km, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Pascal's principle: a pressure increase applied to an enclosed fluid at rest reaches every point of that fluid, and every wall confining it, at full strength. Use the supplied topic model Y_out = s/r^2, where Y_out is OUTPUT RISE d₂ (m). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. r is Piston diameter ratio d₂/d₁, measured in ×. Keep s (Input piston stroke d₁) = 0.35 m fixed. For this exercise use only 1.61688 × ≤ r ≤ 2.48313 × on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.076049619 m, with a bounded reading uncertainty ±0.0028929 m. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000086625 ×, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.