Formula solvers starting with I — 46 solvers

IBU from Utilisation

IBU = \frac{U \times m_\alpha}{V}

Brewing & FermentationThe bitterness a hop addition delivers: the mass of alpha acids that isomerised, divided by the volume it ended up in. One IBU is one milligram of iso-alpha acid per litre.

Ideal Gas Law

P V = n R T

ChemistryPhysicsRelates pressure, volume, amount, and temperature of a gas. R = 8.314 J/(mol·K).

Ideal Hover Power (Momentum Theory)

P = \frac{T^{3/2}}{\sqrt{2 \rho A}}

Aerospace & FlightThe least power any device could use to hold a given thrust up on a given disc: thrust times the induced velocity it makes. Power goes as thrust to the THREE-HALVES power, which is why a helicopter cannot simply be scaled up.

IDF Rainfall Intensity (Three-Parameter)

i = \frac{a}{(t + b)^{c}}

Water & WastewaterCivil & SurveyingFluid MechanicsThe general intensity-duration-frequency form: design rainfall intensity from the storm duration, using the three constants a, b and c fitted to a local rain gauge record for one return period. The constants are regional and unit-bound — a is stated here for intensity in mm/h against duration in minutes — and the duration you type is converted to minutes before the fit is applied.

Image Count for a Block

n = \left( \left\lfloor \frac{L}{B} \right\rfloor + 1 \right) \left( \left\lfloor \frac{W}{S} \right\rfloor + 1 \right)

Photogrammetry & Remote SensingHow many frames a rectangular block costs: the frames along one line, times the number of lines. Each count is a division that rounds down plus one, because the first exposure sits at the start of the line and the first line sits at the edge of the block. This is the number that decides battery swaps, card space and processing time.

Image Ground Footprint

A = \mathrm{GSD}^{2} \, n_x \, n_y

Photogrammetry & Remote SensingWhat one frame covers on the ground: the ground sample distance multiplied by the pixel count, in each direction. The area follows as the product of the two edges. This is the number that turns a GSD into a flight plan — it sets the air base, the line spacing and, through those, the whole image count.

Impedance Contrast Amplification

A = \sqrt{\dfrac{\rho_1 V_{s1}}{\rho_2 V_{s2}}}

SeismologyHow much a shear wave grows when it passes from stiff rock into soft soil: the square root of the impedance ratio between the two. The conservation-of-energy reason that soft ground shakes harder than the rock beside it.

Impulse (J = FΔt)

J = F \, \Delta t

MechanicsPhysicsImpulse delivered by an average force acting over a contact time, equal to the change in momentum.

Inbreeding Coefficient after Selfing

F = 1 - \left(\tfrac{1}{2}\right)^{n}

Plant Breeding & GeneticsHow homozygous a line has become after a number of self-pollinated generations. Halving the remaining heterozygosity each generation is what turns a cross into a fixed line.

Incidence Rate (Person-Time)

\mathrm{IR} = 1000 \cdot \frac{C}{\mathrm{PT}}

Epidemiology & Diagnostic AccuracyNew cases divided by the person-time at risk, reported here per 1,000 person-years. The denominator counts time rather than people, so someone followed for two years contributes twice what someone followed for one does.

Index of Refraction (n = c/v)

n = \frac{c}{v}

OpticsPhysicsHow much a medium slows light: the ratio of c to the speed of light in the medium.

Induced Drag Coefficient

C_{Di} = \frac{C_L^{2}}{\pi \, AR \, e}

Aerospace & FlightThe drag a wing cannot avoid paying for the lift it makes, from Prandtl's lifting-line theory: lift coefficient squared over π times aspect ratio times the span efficiency factor. It grows with the SQUARE of C_L, which is why it dominates at low speed.

Induced Velocity in Hover (Momentum Theory)

v_i = \sqrt{\frac{T}{2 \rho A}}

Aerospace & FlightHow fast a hovering rotor drives air down through its own disc, from the Rankine–Froude actuator disc: the square root of disc loading over twice the air density. The far wake moves at twice this speed, and the disc contracts to half its area to carry it.

Induction Motor Slip

s = \frac{100 \, (N_{s} - N_{r})}{N_{s}}

Electrical TradeElectricity & MagnetismPercent slip of an induction motor: how far the rotor falls behind the rotating magnetic field, as a share of synchronous speed.

Inductive Reactance (X_L = 2πfL)

X_L = 2\pi f L

Electricity & MagnetismPhysicsAn inductor's opposition to AC current, rising in proportion to frequency.

Infinite Geometric Series

S = \frac{a_1}{1 - r}

AlgebraLimit of a geometric series with ratio between −1 and 1, summed forever.

Infinite Slope Factor of Safety with Cohesion and Pore Pressure

FS = \frac{c' + \left(\gamma z\cos^{2}\beta - u\right)\tan\phi'}{\gamma z\sin\beta\cos\beta}

Soil MechanicsMechanicsThe general infinite-slope factor of safety: Mohr-Coulomb strength on a slope-parallel plane at depth z, with cohesion in the numerator and the pore pressure eating the effective normal stress that friction acts on.

Infinite Slope Factor of Safety with Slope-Parallel Seepage

FS = \frac{\gamma'}{\gamma_{sat}}\cdot\frac{\tan\phi'}{\tan\beta}

Soil MechanicsFluid MechanicsFactor of safety of a long shallow slide in cohesionless soil with the water table at the ground surface and seepage running parallel to the slope — the wet-season case, worse than the dry one by the ratio of buoyant to saturated unit weight.

Infinite Slope Factor of Safety, Dry Cohesionless Soil

FS = \frac{\tan\phi'}{\tan\beta}

Soil MechanicsMechanicsFactor of safety of a long shallow slide in dry cohesionless soil, where the whole answer is the ratio of two tangents: the friction angle over the slope angle. Depth and unit weight cancel out completely.

Inflation Rate from a Price Index

\pi = \frac{I_2 - I_1}{I_1}

Prices & MarketsPercentage change in a price index between two periods — the ordinary definition of an inflation rate. Divide by the EARLIER index, always, because the rate is measured against where prices started.

Infusion Volume for a Mash Step

W = \frac{(T_2 - T_1)\,(0.2\,G + W_m)}{T_w - T_2}

Brewing & FermentationHow much hot water to add to a mash already sitting at one temperature to bring it to the next rest. The heat balance behind every step-mash schedule.

Initial Great Circle Bearing

\theta = \operatorname{atan2}\!\left(\sin\Delta\lambda\,\cos\varphi_2,\; \cos\varphi_1\sin\varphi_2 - \sin\varphi_1\cos\varphi_2\cos\Delta\lambda\right)

Navigation & PositionThe course at the moment of departure along the great circle to a destination, measured clockwise from true north. It is the starting bearing only — on any track but a meridian or the equator it changes the whole way.

Injection Moulding Clamp Force

F = p_{cav} \, A_{proj}

Polymers & Plastics ProcessingPressure inside the cavity acting on the area the parting line sees, which is the force trying to blow the mould open. It decides which press the job goes on, and it is the single most common reason a tool flashes: the cavity pressure was underestimated, or the projected area forgot the runner.

Injection Moulding Cooling Time

t = \frac{h^{2}}{\pi^{2}\alpha}\, \ln\!\left[\frac{4}{\pi}\cdot\frac{T_m - T_w}{T_e - T_w}\right]

Polymers & Plastics ProcessingThe one-term solution to transient conduction in a slab cooled from both faces, written the way a moulder needs it: how long the part must sit in the tool before it is stiff enough to eject. The wall thickness is SQUARED, which is why it dominates every other term on the page and why halving the wall quarters the cooling time.

Inside Swing Amplitude

A_i = f \, A_o

Heat TransferThe decrement factor turned into degrees. Multiply the outdoor swing by the fraction that survives the wall and you have the swing at the inside face — the number a room's occupant actually experiences, as opposed to a dimensionless ratio nobody can feel.

Instantaneous Selectivity

S_{D/U} = \frac{r_D}{r_U}

Chemical EngineeringThe rate of the reaction you want divided by the rate of the one you do not, at the conditions inside the reactor right now. When two reactions compete for the same feed, this ratio decides which vessel to build and where to run it.

Interphase Mass Flux

N_A = K_y \left( y - y^{*} \right)

Chemical EngineeringThe point of the whole apparatus: a coefficient multiplied by a departure from equilibrium gives the moles crossing each square metre of interface every second. Everything else in mass transfer exists to estimate one of the two terms on the right.

Interquartile Range (IQR)

IQR = Q_3 - Q_1

StatisticsAlgebraThe width of the middle half of a data set, from the first quartile to the third, and the spread measure box plots are built on.

Inventory Turnover Ratio

T = \frac{\mathit{COGS}}{\bar{I}}

Industrial EngineeringMoney & BusinessHow many times a year the stockroom empties and refills: cost of goods sold divided by the average value of inventory on hand.

Inverse of a 2×2 Matrix (Top-Left Element)

(M^{-1})_{11} = \frac{d}{ad - bc}

Vectors & MatricesAlgebraGives the top-left entry of the inverse of a two-by-two matrix: the opposite diagonal entry divided by the determinant.

Inverse Rule of Mixtures — Transverse Modulus

\frac{1}{E_2} = \frac{V_f}{E_f} + \frac{1 - V_f}{E_m}

Strength of MaterialsStiffness across the fibres, where the two phases share the same stress instead of the same strain and their compliances add. The answer is dominated by whichever phase is softer, which for a fibre composite is always the resin — and that is why the transverse modulus of a lamina is a small fraction of the longitudinal one rather than a modest reduction from it.

Inverse Variation

y = \frac{k}{x}

AlgebraRelation in which y falls as x rises so that their product stays at a constant k, the shape of Boyle's law and gear ratios.

Inverse-Square Law for Sound

I = \frac{P}{4\pi r^{2}}

Waves & OscillationsPhysicsAcoustics & NoiseA point source's intensity falls with the square of distance as its power spreads over an expanding sphere.

Inverting Op-Amp Gain

G = -\frac{R_{f}}{R_{in}}

Electrical TradeElectricity & MagnetismClosed-loop gain of an inverting operational amplifier, set purely by the ratio of feedback resistor to input resistor.

Ion Concentration as CaCO₃ Equivalent

C_{\mathrm{CaCO_3}} = C_{\mathrm{ion}} \times \frac{50.04}{\mathrm{EW}}

Water TreatmentChemistryRestates any ion's concentration in the trade's common currency of CaCO₃ by scaling with the ratio of equivalent weights, 50.04 over the ion's own.

Iribarren Number (Surf-Similarity Parameter)

\xi = \frac{\tan\beta}{\sqrt{H / L_0}}

Coastal & Ocean WavesThe beach slope divided by the square root of the wave steepness. Ramón Iribarren and Casto Nogales published it in 1949, and it turns out to classify almost everything about a surf zone: whether a wave spills, plunges, collapses or surges, how much of it runs up, how much reflects, and how armour on a slope behaves.

Irrigation Application Efficiency

E_a = \frac{W_s}{W_d}

Crop ProductionThe share of the water delivered to a field that ends up stored in the root zone where the crop can use it. What is missing went to runoff, to wind and evaporation, or straight past the roots to the water table.

Irrigation Interval

I = \frac{RAW}{ET_c}

Crop ProductionHow many days a crop can go between irrigations: the depth of water it is allowed to use, divided by the depth it uses each day. The whole of irrigation scheduling in one division.

Irrigation Set Run Time

t = \frac{d \, A}{Q}

Crop ProductionHow long a set must run to lay down a wanted depth of water over a known area at a known flow rate — depth times area is a volume, and a volume divided by a flow rate is a time.

Irrigation System Capacity

Q = \frac{A \, ET_p}{E_a \, f}

Crop ProductionThe flow an irrigation system must deliver to keep up with peak crop water use, once application efficiency and the hours a day the system can actually run are accounted for. The sizing calculation behind every pump, well and mainline on the farm.

Irwin Plastic Zone Size

r_p = \dfrac{1}{\beta \pi} \left( \dfrac{K}{\sigma_y} \right)^{2}

Strength of MaterialsThe elastic solution says the stress at a crack tip is infinite, which no material tolerates: it yields instead, over a small zone whose size this estimates. The zone has to be small compared with the crack and the remaining ligament, or linear elastic fracture mechanics does not apply at all — which makes this the validity check for every other page in this set.

Isentropic Pressure Ratio

\frac{p_0}{p} = \left(1 + \frac{\gamma - 1}{2} M^{2}\right)^{\frac{\gamma}{\gamma - 1}}

Aerospace & FlightThe pressure a moving gas would reach if brought smoothly to rest, over the pressure it has while moving. The same physics as the temperature ratio, raised to γ/(γ−1) — and the relation that sets a nozzle's exit pressure.

Isentropic Temperature Ratio

\frac{T_0}{T} = 1 + \frac{\gamma - 1}{2} M^{2}

Aerospace & FlightHow much hotter a gas is when it is brought to rest than when it is moving: the stagnation temperature over the static temperature, set entirely by Mach number and the ratio of specific heats.

Isokinetic Sampling Rate

Q_n = v_s A_n

Air Quality & DispersionThe flow a sampling train must draw so that gas enters the nozzle at exactly the stack velocity. Get this wrong and the particulate result is biased in a direction you can predict from the sign of the error.

IV Drip Rate (Drops per Minute)

R_{gtt} = \frac{V \cdot f}{t}

Biomedical & ClinicalDrops per minute for a gravity-fed IV set: the volume, multiplied by the tubing's drop factor, divided by the time.

IV Infusion Rate (mL per Hour)

R = \frac{V}{t}

Biomedical & ClinicalEveryday & HealthThe rate to set on a volumetric infusion pump: the volume to be infused divided by the time it should take.