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Toth (1971) 3-Parameter Adsorption Isotherm Formula: Henry’s Law & Monolayer Limit
Model sub-monolayer gas adsorption (CO2 capture, CH4 shale/coalbed storage, VOC zeolites) using József Tóth’s (1971) thermodynamically consistent 3-parameter isotherm q = q_max * b * P / (1 + (b * P)^t)^(1/t).
Read the guide →Contract Theory & Corporate FinanceTownsend Costly State Verification Debt Formula
Solve Robert Townsend’s (1979) optimal standard debt face value D* and expected bankruptcy audit cost K*(D*/Y_max) under costly state verification.
Read the guide →Engineering & ScienceTownsend Electron Avalanche Current & Spark Breakdown Criterion Formula
Calculate pre-breakdown Townsend gas-discharge current I = (I0 * EXP(alpha * d)) / (1 - gamma * (EXP(alpha * d) - 1)) and test the Townsend self-sustaining spark criterion gamma * (EXP(alpha * d) - 1) >= 1.
Read the guide →Spreadsheet Learning & Team TrainingTrack Server-Verified Academy Lesson Progress vs AI Hint Practice
Audit team spreadsheet training by separating local draft attempts and AI hint sessions from server-verified Academy lesson completions in Excel and Arte Sheets.
Read the guide →LogicalTraining completion by department as of a selected date
Classify each assignment as completed on time, completed late, overdue or in progress at a chosen reporting date, then calculate department completion rates.
Read the guide →Math & TrigTraining, CPE, and volunteer hours requirement tracker (remaining & weekly pace)
Track a chosen hours target, capped progress and the average weekly pace needed over the remaining time.
Read the guide →Electrical Power Systems & Energy EngineeringTransformer All-Day Efficiency: Core vs Copper Loss Formula
Calculate distribution transformer peak-efficiency load fraction x_opt = SQRT(P_core / P_cu,fl), peak instantaneous efficiency, and 24-hour duty-cycle All-Day Energy Efficiency.
Read the guide →Substation Power Transformers & Dissolved Gas Protection, Conservator & Buchholz RelayTransformer Conservator Expansion & Buchholz Gas Formula
Calculate Mineral-Oil Seasonal/Load Volume Swing dV (L), Required Overhead Conservator Tank Capacity (L and % of Main Tank), and Buchholz Relay Gas Evolution Rate (mL/h).
Read the guide →Substation Protection & Power Systems Engineering, Blume-Specht Transformer InrushTransformer Magnetizing Inrush Peak & Joule I²t Formula
Calculate Rated Primary RMS Current (A), Worst-Case Zero-Crossing Peak Inrush Multiple (pu), First-Cycle Peak Inrush Current (kA), and Exponential-Decay Joule Integral I²t (kA²·s).
Read the guide →High-Voltage Power Systems, EHV AC Transmission Lines & Reactive CompensationTransmission Line SIL & Ferranti Voltage Rise Formula
Calculate Characteristic Surge Impedance Z_c (Ohms), Surge Impedance Loading SIL (MW), Open-Circuit Receiving-End Ferranti Voltage V_r (kV), and Overvoltage Percentage (%).
Read the guide →Dynamic ArraysTranspose a table without zeros: suppress blank-to-0 coercion with LET & IF
Extract a coordinate from a transposed numeric grid and display a blank string for either an empty source cell or an explicit zero.
Read the guide →Lookup & ReferenceTRANSPOSE function: rotate horizontal tables into vertical columns live
Flip a wide horizontal table into a vertical column layout with TRANSPOSE while keeping live formula links to the source cells.
Read the guide →Healthcare & ClinicalTranstubular Potassium Gradient (TTKG) & Urine Anion Gap (UAG)
Calculate aldosterone-mediated Transtubular Potassium Gradient TTKG = (U_K / P_K) / (U_Osm / P_Osm) (valid when U_Osm > P_Osm) and Urine Anion Gap UAG = U_Na + U_K - U_Cl for renal tubular acidosis.
Read the guide →Statistical & Mathematical FormulasTrapezoidal Rule Area Under the Curve (AUC) With SUMPRODUCT
Integrate discrete sensor, pharmacokinetic (AUC), or power-load curves across uneven X-intervals without helper columns using 0.5 * SUMPRODUCT(A3:A6 - A2:A5, B2:B5 + B3:B6).
Read the guide →Quantitative FinanceTreasury 32nds Fractional Bond Quote Conversion (DOLLARDE & DOLLARFR Formula)
Convert US Treasury fractional handle-and-ticks quotes (such as 99.16 = 99 and 16/32 = 99.50) to exact decimal prices and back using closed-form INT, LEN, and power-of-10 arithmetic.
Read the guide →Quantitative FinanceTreasury Bill Bank Discount Rate (DISC) & Bond Equivalent Yield Formula
Calculate the 360-day money-market Bank Discount Rate (DISC) alongside the 365-day Bond Equivalent Yield (BEY) to compare zero-coupon T-bills fairly against coupon-bearing notes.
Read the guide →Fixed Income Trading & Cash-Futures BasisTreasury Bond Futures Gross Basis & BNOC Formula
Calculate Deliverable Bond Gross Price Basis (Clean_Price - Futures * CF), Net Coupon-Minus-Repo Carry, Basis Net of Carry (BNOC), and Cheapest-to-Deliver (CTD) Cash-and-Carry Arbitrage Status.
Read the guide →Financial Statement Analysis & Equity Valuation (ASC 260)Treasury Stock & If-Converted Diluted EPS Formula
Calculate Basic EPS, Treasury-Stock-Method Incremental Option Shares MAX(0, N_opt*(1 - Strike/Avg_Price)), and ASC 260 / IAS 33 Antidilution-Guarded Diluted EPS across Convertible Debt.
Read the guide →Charts & Visual DataTreemap Chart Formula: Proportional Tile Area (px^2) & Squarified Aspect Ratio
Calculate exact rectangular tile pixel areas, strip thicknesses, and Bruls-Huizing-van Wijk Squarified Aspect Ratios MAX(w/h, h/w) for hierarchical Treemap Charts without negative-value distortions.
Read the guide →FinancialTreynor-Black Model formulas: Appraisal Ratio & Optimal Active Sleeve Weight
Explore the model-implied active weight, appraisal ratio and theoretical Sharpe ratio under a single-index portfolio model.
Read the guide →FinancialTrigger Fixed Payment Every N Columns or Months (MOD + COLUMN)
Place a fixed amount every N equal-width period columns, choosing either the first or last column of each cycle. Includes interval validation and a six-period experiment.
Read the guide →Healthcare & Clinical CalculationsTriglyceride-Glucose (TyG) Index & TyG-BMI Insulin Resistance Formula
Screen for insulin resistance and MASLD without fasting insulin assays using Simental-Mendía’s logarithmic TyG = LN((Fasting_TG_mgdL * Fasting_Glucose_mgdL) / 2) and composite TyG-BMI index.
Read the guide →Cleaning imported dataTRIM in Excel: clean extra spaces without joining words
Clean inconsistent customer labels while preserving the space between words. Then handle a nonbreaking space copied from a web page and check what the cleanup changes.
Read the guide →TextTrim text to the first N words (or extract the last N words)
Shorten long product titles and descriptions cleanly at word boundaries using TEXTBEFORE(..., " ", N) without slicing words mid-syllable.
Read the guide →Clinical Medicine & PhysiologyTRISS Trauma Survival Probability Formula: Blunt vs Penetrating RTS + ISS + Age Logit
Compute Carl R. Boyd et al.’s (1987, MTOS) Trauma and Injury Severity Score (TRISS) hospital survival probability P_s = 1 / (1 + EXP(-b)) combining physiological RTS, anatomical ISS, Age >= 55, and Blunt vs Penetrating trauma coefficients.
Read the guide →Engineering & ScienceTrouton’s Rule vs Kistiakowsky Equation: Boiling ΔHvap & ΔSvap
Estimate normal-boiling enthalpy of vaporization ΔH_vap,b (kJ/mol) from normal boiling temperature T_b (K) alone using George Kistiakowsky’s logarithmic entropy law ΔS_vap,b = 36.61 + R * LN(T_b) vs Frederick Trouton’s 88 J/(mol·K) constant.
Read the guide →StatisticalTrue Shooting Percentage (TS%) & Effective FG% (eFG%) Basketball formulas
Calculate effective field-goal percentage and the conventional box-score approximation of true shooting, with consistent scoring inputs.
Read the guide →Healthcare & ClinicalTruelove and Witts Criteria for Acute Severe Ulcerative Colitis (ASUC)
Classify ulcerative colitis flares into Mild, Moderate, or Acute Severe UC (ASUC: >= 6 bloody stools/day plus >= 1 systemic toxicity marker: T > 37.8 °C, HR > 90, Hgb < 10.5 g/dL, or CRP/ESR > 30).
Read the guide →MathTRUNC vs INT: chop decimal places without rounding and handle negative values
Understand why TRUNC(-4.89) gives -4 while INT(-4.89) gives -5, and use TRUNC's second argument to cut at N decimal places.
Read the guide →Engineering & ScienceTsai-Hill Orthotropic Composite Ply Failure Index & Strength Ratio
Evaluate biaxial + shear lamina failure under combined in-plane stresses (sigma_1, sigma_2, tau_12) using the quadratic Tsai-Hill failure index FI and reserve strength factor SF = 1 / SQRT(FI).
Read the guide →Engineering & ScienceTsai-Wu Tensor Composite Failure Criterion (Tension vs Compression Asymmetry)
Calculate the Tsai-Wu orthotropic lamina failure index FI = F1*s1 + F2*s2 + F11*s1^2 + F22*s2^2 + F66*t12^2 + 2*F12*s1*s2 accounting for unequal tensile (Xt, Yt) and compressive (Xc, Yc) composite strengths.
Read the guide →HealthcareTSAT and Ganzoni Iron Deficit: Spreadsheet Arithmetic and Units
Calculate transferrin saturation and a Ganzoni iron estimate using explicit units, supplied target hemoglobin and storage iron, with missing-input checks.
Read the guide →Survey Sampling & Experimental DesignTschuprow-Neyman Optimal Stratified Allocation Formula
Calculate Cost-Adjusted Optimal Stratified Sample Sizes n_h* proportional to W_h * S_h / SQRT(c_h), Minimum Estimator Variance under fixed budget B_tot, and Variance Reduction vs Proportional Allocation.
Read the guide →EngineeringTsiolkovsky Ideal Rocket Equation formula: Delta-v = Isp * g0 * LN(m0 / mf)
Calculate ideal delta-v from initial and final mass at constant specific impulse, and invert the model for a propellant fraction.
Read the guide →Naval Architecture & Towage Engineering, ASD Tugboat Static Bollard Pull & Ducted NozzlesTugboat Static Bollard Pull & Ducted Nozzle Thrust Formula
Calculate Total Swept Propeller Disk Area (m²), Certified Continuous Bollard Pull (tonnes), Specific Thrust Efficiency (kg/kW), and Storm Towline Reserve Margin (%).
Read the guide →Math & StatisticalTukey HSD & Tukey-Kramer Post-Hoc ANOVA Pairwise Test Formula
Calculate Tukey Honestly Significant Difference HSD = q_crit * SQRT(MS_W / n) for balanced groups and the Tukey-Kramer Studentized Range statistic q = ABS(M1 - M2) / SQRT((MS_W / 2) * (1/n1 + 1/n2)) for unequal sample sizes.
Read the guide →Public Choice & Game TheoryTullock Rent-Seeking Contest Effort & Dissipation Formula
Calculate symmetric N-player Tullock contest expenditure x* = ((N - 1)*r / N^2)*V, aggregate rent dissipation ((N - 1)/N)*r, and asymmetric two-rival lottery outlays.
Read the guide →Aerospace Propulsion, Gas Turbines & Flight ThermodynamicsTurbofan TSFC, Propulsive Efficiency & Overall Efficiency Formula
Calculate High-Bypass Turbofan Froude Propulsive Efficiency 2 / (1 + Vj / V0), Hourly Fuel Burn from Thrust-Specific Fuel Consumption (TSFC), Propulsive Thrust Power F*V0, and Overall Aircraft Thermodynamic Efficiency.
Read the guide →Steam & Gas Turbines, Centrifugal Compressors & Martin Labyrinth Seal LeakageTurbomachinery Labyrinth Seal Martin Leakage Formula
Calculate Annular Tooth-Tip Clearance Area (mm²), Martin Compressible Expansion-Throttling Factor, Parasitic Gas Leakage Mass Flow (kg/s), and Hourly Leakage Loss (kg/h).
Read the guide →High-Vacuum & Mass Spectrometry Engineering, Kruger-Shapiro Turbomolecular Pump PhysicsTurbomolecular Pump Compression Ratio & H2 Back-Streaming
Calculate Rotor Blade Linear Speed vb (m/s), Maxwell Most-Probable Molecular Speed vm (m/s), Blade-to-Molecular Speed Ratio, and Log10 Zero-Flow Compression Ratio log10(K0).
Read the guide →Civil, Structural & Geotechnical EngineeringTurc (1954) & Budyko Curve Watershed Water-Balance Formula: Annual AET & Runoff Yield
Partition long-term annual catchment precipitation P into Actual Evapotranspiration (AET) and Streamflow Runoff Q = P - AET using Mikhail I. Budyko’s (1974) Aridity Curve (P, PET) and Lucien Turc’s (1954) Temperature-Precipitation Law.
Read the guide →Statistical AnalysisTwo-Dice Sum Probability Distribution, Tail Thresholds & Expected Payoff Formulas
Calculate exact 2d6 discrete convolution ways 6 - ABS(Sum - 7), closed-form tail probabilities P(Sum >= Target), and expected game value without volatile RANDARRAY recalculation.
Read the guide →Lookup & ReferenceTwo-Dimensional VLOOKUP + MATCH Formula: Dynamic Row & Column Matrix Lookup
Replace brittle hardcoded VLOOKUP column numbers with MATCH(ColHeader, A1:D1, 0) or INDEX + MATCH + MATCH to query any row-and-column intersection dynamically.
Read the guide →ASME B30.9 Overhead Crane Rigging, Wire-Rope/Chain Bridle Slings & Spreader BeamsTwo-Leg Bridle Sling Angle Tension & WLL Formula
Calculate Per-Leg Sling Angle Multiplier 1/(2*SIN(theta)), Actual Tension Per Sling Leg (t), Inward Horizontal Crushing Force on Load (t), and Derated Hitch Working Load Limit (t).
Read the guide →StatisticalTwo-level Wells DVT score: a complete binary-flag spreadsheet exercise
Practice SUM, IF and input validation with all nine positive DVT criteria and a separate alternative-diagnosis deduction using fictional data.
Read the guide →Managerial Economics & Nonlinear PricingTwo-Part Tariff Access Fee & Usage Price Formula (Oi Rule)
Optimize a subscription + usage two-part tariff (T*, p*) across homogeneous and heterogeneous customer segments using Walter Oi’s (1971) Disneyland Dilemma formula.
Read the guide →Mineral Processing, Concentrator Mass Balance & Froth FlotationTwo-Product Froth Flotation Mass Pull & Metal Recovery Formula
Calculate Concentrate Mass Pull C/F (%), Metallurgical Metal Recovery R (%), Ratio of Concentration K, and Hourly Concentrate Filter-Cake Tonnage (t/hr) from Feed, Concentrate & Tailings Assays.
Read the guide →StatisticalTwo-sample t-test formula (Welch's unequal-variance t-statistic)
Compare two independent group means (e.g., A/B order value or cycle time) using Welch's t-statistic (Mean1 - Mean2) / SQRT(Var1/sampleCountA + Var2/sampleCountB).
Read the guide →Platform Economics & Digital MarketplacesTwo-Sided Platform Cross-Subsidy Pricing Formula
Solve Armstrong / Rochet-Tirole two-sided platform prices p_1* = c_1 + n_1* - alpha_2 * n_2* and explain why optimal consumer prices often sit below marginal cost.
Read the guide →Lookup & ReferenceTwo-way approximate match lookup: 2D bracket grid (INDEX & XMATCH)
Look up rates in a 2D bracket matrix (such as Order Weight × Credit Tier or LTV × FICO Score) where both row and column headers are numeric thresholds.
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