Kyle Standing · YouTube · 1:48
This short whiteboard-style lesson demonstrates linear interpolation using a thermodynamic property table for compressed liquid water. After showing the table at bar = 2.5 MPa, the presenter chooses the rows °C with /kg and °C with /kg, then estimates v at the intermediate temperature °C. A straight-line graph converts the table data into a proportionality statement, giving (220 - 200)/(212 - 200) = (1.1898 - 1.1555)/(x - 1.1555). The equation is rearranged to , evaluated as , and checked by confirming that the result lies between the two known specific volumes.
Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.
Generated from the video's visuals and explanation; not verbatim speech.
The clip opens by naming the goal: to estimate an unknown intermediate number by linear interpolation. The title card and the speaker’s first sentence frame the lesson as a practical shortcut rather than a formal proof.
A compressed liquid water property table supplies the data. The relevant fixed condition is bar = 2.5 MPa, and the two rows used later are °C with /kg and °C with /kg. The speaker contrasts an easy direct lookup at 200°C with the harder case of a temperature not printed in the table.
The target input is set to 212°C. Although 210 is mentioned briefly in speech, the worked example immediately switches to 212, and all subsequent graphics and formulas use 212 consistently.
The table values are translated into a graph. Temperature is placed on the vertical axis and specific volume on the horizontal axis. The known endpoints become (1.1555, 200) and (1.1898, 220), while the desired unknown is an x-value on the horizontal axis corresponding to 212 on the vertical axis. A straight green line through the endpoints encodes the linear assumption.
From that geometry, the video writes the proportion (220 - 200)/(212 - 200) = (1.1898 - 1.1555)/(x - 1.1555). The spoken mnemonic is that each side uses top minus bottom over middle minus bottom. Mathematically, this says the fractional position of 212 within the temperature interval matches the fractional position of x within the specific-volume interval.
The equation is then rearranged to isolate the unknown: . This form makes the structure clear: start from the lower known v-value and add the portion of the total v-change corresponding to how far 212 lies between 200 and 220.
Evaluating the expression gives . The presenter checks the answer by asking whether it lies between the two known specific volumes. Since , the result passes the video’s simple sanity check and the example concludes.
Linear interpolation estimates an unknown intermediate value by assuming the relation between two variables is a straight line over a small interval. In this video, the known data come from adjacent rows of a thermodynamic table, and the unknown value is read off the line joining those two rows.
The worked problem uses the rows °C, /kg and °C, /kg at bar = 2.5 MPa. The target temperature is 212°C, which is not listed directly, so the specific volume must be estimated.
The graph places temperature on the vertical axis and specific volume on the horizontal axis. The two known table rows define endpoints of a green straight segment, and the target 212°C determines a point on that segment whose horizontal coordinate is the unknown x.
The video writes (220 - 200)/(212 - 200) = (1.1898 - 1.1555)/(x - 1.1555). Each side compares a full interval to a partial interval measured from the same lower endpoint, expressing constant slope between the known points and the unknown point.
By cross-multiplying and isolating x, the formula becomes . This says: take the lower known v-value and add the fraction of the total v-change corresponding to the target’s position in the temperature interval.
Substituting the numbers gives /kg. The video checks plausibility by confirming that this value lies between the bracketing specific volumes 1.1555 and 1.1898.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
Title card reads "HOW TO LINEAR INTERPOLATE !!".
Speaker says he will show how to linear interpolate between numbers.
linear interpolate
The method being taught: estimating an intermediate value from two known endpoints by assuming a straight-line relationship.
Applied numerical estimation using tabulated data.
Table header shows T with unit °C.
Graph vertical axis is labeled T and marked 200, 212, and 220.
Speaker says temperature is on this axis.
T
Temperature variable used as the independent coordinate in the interpolation example.
°C; in the worked example the relevant values are 200, 212, and 220.
Table header shows with unit /kg.
Graph horizontal axis is labeled V and marked 1.1555, x, and 1.1898.
Speaker says the other axis has the specific volume and corresponding values.
The graph label uses capital V while the table uses lowercase v; the video does not explicitly comment on this notation difference.
v
Specific volume of compressed liquid water, treated as the dependent quantity in the interpolation example.
/kg; displayed table entries are scaled by in the header, but the plotted numeric labels 1.1555 and 1.1898 match the table row values directly.
Table includes a column labeled u with unit kJ/kg.
This column is visible but not used in the worked interpolation.
u
Internal energy column in the thermodynamic table.
kJ/kg.
Table condition line reads bar = 2.5 MPa.
p
Pressure condition under which the tabulated properties are listed.
25 bar = 2.5 MPa.
Table condition line includes (T_sat = 223.99°C).
T_sat
Saturation temperature associated with the listed pressure condition.
223.99°C.
Graph marks 220 on the T-axis and 1.1898 on the V-axis as the upper-right endpoint.
Speaker says 220 is chucked at the top and identifies the corresponding values.
top number
The larger of the two known bracketing values on a given axis; here 220 for T and 1.1898 for v.
Used in the numerator differences of the interpolation formula.
Graph marks 200 on the T-axis and 1.1555 on the V-axis as the lower-left endpoint.
Speaker says 200 is chucked at the bottom and repeatedly subtracts the bottom number.
bottom number
The smaller of the two known bracketing values on a given axis; here 200 for T and 1.1555 for v.
Subtracted from both the top and middle values in the formula.
The number 212 appears before the graph and later on the T-axis as the target temperature.
Speaker says they ask at 212, then changes to let's say 212, and says 212 is what we want to find.
middle number
The known target input value lying between the two bracketing endpoints; here 212 on the T-axis.
Appears in the denominator on the T side of the proportion.
Graph horizontal axis contains an unknown point labeled x between 1.1555 and 1.1898.
Speaker says there is x that we want to find and later says now we just need to solve for x.
x
The unknown interpolated specific volume corresponding to .
Real-valued intermediate quantity between 1.1555 and 1.1898.
Opening statement: today I'm going to show you how to linear interpolate between numbers.
A straight green segment connects the two known table points, and the target point lies on that segment.
Displayed proportion equates differences on the T-axis to differences on the v-axis.
The video teaches linear interpolation as a way to estimate an unknown intermediate value when the target input lies between two known rows of a table. It assumes the relation between the two variables over that small interval is a straight line, so proportional changes in one coordinate match proportional changes in the other.
Two known bracketing data points are available.
The target input lies between those two known inputs.
The relationship over the interval is approximated as linear.
On-screen equation: (220 - 200)/(212 - 200) = (1.1898 - 1.1555)/(x - 1.1555).
Speaker describes the rule as top number minus bottom number divided by middle number minus bottom number on both sides.
The displayed formula states that the ratio of the full interval length to the partial interval length is the same on both axes. This is equivalent to saying the slope of the line through the two known points equals the slope from the lower known point to the unknown point.
The three T-values are ordered as .
The corresponding v-values are ordered consistently on the line.
Denominators are nonzero.
On-screen rearranged expression: .
Speaker says you just got to do algebra jiggy pokery and get x equals something.
Starting from the equal-ratio equation, the video cross-multiplies and solves for x. The resulting expression adds the lower known v-value to the fraction of the total v-increment corresponding to the fractional position of the target T within the T-interval.
Use the same lower endpoint 1.1555 that appears in the original denominator.
The formula assumes the linear model shown in the graph.
Final displayed result: .
Speaker asks whether it is in between the values and answers yes it is.
The point x is drawn between 1.1555 and 1.1898 on the horizontal axis.
After computing x, the video verifies plausibility by checking that the result lies between the two known specific volumes. This is a simple consistency test for interpolation inside the bracketing interval.
The target input must be strictly between the two known inputs.
The two known outputs must bound the interpolated output on the assumed line.
A single straight green line is drawn through the two known points and the unknown point.
The lesson title and spoken explanation identify the procedure as linear interpolation.
The video does not discuss error bounds or when the straight-line approximation is inaccurate.
For the purpose of this example, the relation between T and v between the rows 200°C and 220°C is modeled as a straight line, allowing proportional interpolation.
Two known endpoints are available.
The target temperature lies between them.
The user accepts a linear approximation over the interval.
For the displayed pair of table rows and the target value 212°C in this example.
Speaker says, Is it in between the values? In between these values? Yes it is.
Computed value shown as .
x is placed between 1.1555 and 1.1898 on the axis.
Betweenness is presented as a quick check, not as proof of numerical correctness.
The computed interpolated value is accepted in the video because it lies between the two known specific-volume endpoints.
The arithmetic was performed correctly.
The target lies inside the bracketing interval.
For the specific numerical result in this example.
Initial proportion shown: (220 - 200)/(212 - 200) = (1.1898 - 1.1555)/(x - 1.1555).
Rearranged expression shown: .
Speaker says now we just need to solve for x and refers to doing algebra.
Intermediate algebraic steps such as explicit cross-multiplication are not written out on screen.
Start from the displayed proportion relating the full and partial intervals on both axes.
Observed on-screen formula.
Cross-multiply to remove the fractions.
Standard algebraic manipulation of an equality of ratios; the video skips writing this intermediate line.
Divide both sides by 220 - 200 to isolate the term containing x.
Algebraic isolation of x; implied by the speaker's statement that one must solve for x.
Add 1.1555 to both sides to obtain x alone.
Matches the final rearranged expression shown on screen.
The unknown interpolated value is expressed as .
Solved symbolic expression is visible before the numeric answer.
Final numeric result shown: .
Speaker says now we just need to plug it into our calculator.
The calculator display itself is not shown; only the typed result appears.
Begin with the rearranged exact expression from the previous derivation.
Observed on-screen formula.
Compute the total change in specific volume across the bracketing rows.
Arithmetic from the displayed numbers.
Compute the distance from the lower known temperature to the target temperature.
Arithmetic from the displayed numbers.
Compute the total temperature interval between the bracketing rows.
Arithmetic from the displayed numbers.
Evaluate the fraction and add it to the lower endpoint to obtain the final interpolated value.
Direct calculation consistent with the final displayed result.
The numerical result shown and verified by calculation is .
Table title reads Compressed Liquid Water with columns T, , and u.
Relevant rows show 200 | 1.1555 | 849.9 and 220 | 1.1898 | 940.7.
Graph plots T vertically and V horizontally with target 212 and unknown x.
Speaker frames this as a common thermodynamics example asking for specific volume at a certain temperature.
The speaker first says 210 and then corrects to 212; the worked graph and formulas use 212.
The table header includes , but the plotted labels use the unscaled row values directly; the video does not explain this scaling presentation.
Given a compressed liquid water property table at bar = 2.5 MPa, estimate the specific volume v at °C using linear interpolation between the tabulated rows at 200°C and 220°C.
°C, /kg.
°C, /kg.
Target temperature T_* = 212°C.
Assumed linear relation between T and v over the interval.
Compute the interpolated unknown specific volume x corresponding to °C.
Set up equal proportional changes using the two known endpoints and the target temperature.
Linear interpolation assumes constant slope between the endpoints.
Solve the proportion for x.
Algebraic rearrangement shown on screen after the speaker says to solve for x.
Evaluate the expression numerically.
Typed final result shown on screen and described as obtained by calculator.
Check that the answer lies between the two known specific volumes.
Explicit verbal and visual sanity check in the video.
/kg
The video verifies the result by confirming that 1.17608 lies between the bracketing values 1.1555 and 1.1898.
Large title text HOW TO LINEAR INTERPOLATE !! appears inside a red hand-drawn border.
Red hand-drawn border
Black uppercase title text
Static title remains on screen for the opening seconds.
No mathematical symbols beyond the topic name are shown yet.
The clip opens by naming the method before any example data appears.
A thermodynamic property table titled Compressed Liquid Water appears on the left.
Visible headers include T (°C), (/kg), u (kJ/kg), and condition bar = 2.5 MPa, (T_sat = 223.99°C).
Some rows are visible but only the 200 and 220 rows are used in the worked example.
Compressed Liquid Water table
Rows for 200 and 220
Cursor pointing near selected rows
The table becomes the source of the endpoint values used later.
The cursor highlights the rows around 200 and 220.
Pressure condition remains fixed at 25 bar = 2.5 MPa.
The table supplies the paired values (200, 1.1555) and (220, 1.1898) that define the interpolation interval.
A large typed number 212 appears to the right of the table.
Speaker says let's say 212 after initially mentioning 210.
The brief earlier mention of 210 is not carried into the worked graph.
Typed numeral 212
Text box border
The target value is introduced before the graph is fully explained.
The chosen target remains 212 in all subsequent formulas.
This establishes the middle input value for the interpolation problem.
A coordinate graph appears with vertical axis T and horizontal axis V.
Marks 200, 212, and 220 appear on the vertical axis; marks 1.1555, x, and 1.1898 appear on the horizontal axis.
A green straight line connects the lower-left and upper-right known points, with guide lines dropping to the axes.
The graph is said to be pre-drawn by the speaker, so its construction is not shown step by step.
Vertical T-axis
Horizontal V-axis
Green straight line
Dashed guide lines
Labels 200, 212, 220, 1.1555, x, 1.1898
Known endpoints are mapped from the table onto a coordinate plane.
The unknown x is positioned between the known v-values.
The line remains straight, encoding the linear assumption.
The target 212 stays between 200 and 220 on the vertical axis.
The picture translates the table rows into a geometric proportional relationship: equal fractional movement along T corresponds to equal fractional movement along v.
First displayed equation: (220 - 200)/(212 - 200) = (1.1898 - 1.1555)/(x - 1.1555).
Second displayed equation: .
Final typed result: .
Red circles and arrows emphasize the paired numbers in the formula and the corresponding graph labels.
The exact order of every annotation stroke is approximate because several marks overlap while the equations remain on screen.
Proportion equation
Rearranged solution equation
Typed result
Red circles and arrows
The visual focus shifts from geometry to algebra.
The unknown x is isolated symbolically and then replaced by a decimal value.
The same endpoint numbers 220, 200, 212, 1.1898, and 1.1555 persist throughout.
The annotations connect each subtraction in the formula to a specific distance on the graph, making the proportionality explicit before the calculator result is shown.
Speaker repeats top number minus bottom number and says it's always minusing this bottom number.
All displayed differences use 200 and 1.1555 as the subtracted lower endpoints.
A learner might mix endpoints inconsistently, for example subtracting the middle value from the top on one side and the bottom from the middle on the other.
The video's rule is to compare both the full interval and the partial interval to the same bottom endpoint: top minus bottom over middle minus bottom on each axis.
The verification question is whether the result is in between the values.
The worked example places 212 strictly between 200 and 220 and x strictly between 1.1555 and 1.1898.
This limitation is inferred from the example setup rather than stated explicitly as a warning.
One might think the same procedure is automatically safe for values outside the tabulated range.
Analysis note: the demonstrated check only makes sense for interpolation inside the bracketing interval; the video does not address extrapolation beyond the known rows.
The graph labels 200, 212, 220 and 1.1555, x, 1.1898 correspond exactly to the numbers in the displayed proportion.
Red circles link formula terms to the matching axis labels.
The geometric straight-line picture is converted into an algebraic equal-ratio formula using the same endpoint and target values.
The proportion equation is followed on screen by the rearranged expression for x.
Speaker says now we just need to solve for x.
The solved expression depends directly on algebraic manipulation of the equal-ratio formula.
Opening statement introduces linear interpolation generally.
The rest of the clip works a specific compressed-water table example.
The general method is demonstrated through a concrete thermodynamic property-table problem.
Numeric result appears after the solved expression.
Speaker checks whether the result is in between the values.
Once x is computed, the video applies a betweenness check to confirm the interpolated value is plausible.
Opening line states the topic is how to linear interpolate between numbers.
Straight-line graph and proportional formula define the method.
Speaker repeatedly says top number minus bottom number and middle number minus bottom number.
Displayed formula uses 200 and 1.1555 as the subtracted endpoints.
Both the proportion and the rearranged expression for x are shown.
Speaker says now we just need to solve for x.
Speaker asks if the result is in between the values and confirms yes.
Final result is compared with 1.1555 and 1.1898.
Table title and headers identify Compressed Liquid Water with v in /kg.
Speaker calls it a common example from thermodynamics asking for specific volume.
Covered · Title card and spoken introduction establish the topic of linear interpolation.
Covered · Blank transition while the speaker continues introducing the lesson; no new mathematical content appears.
Covered · Compressed Liquid Water table appears and the speaker frames the problem as finding specific volume at a temperature not directly listed.
Covered · Target value 212 is typed and the speaker settles on 212 as the example temperature.
Covered · Pre-drawn graph maps the known table rows and the target temperature onto a straight-line geometric model.
Covered · The equal-ratio interpolation formula is displayed and verbally explained as top-minus-bottom over middle-minus-bottom on both axes.
Covered · The formula is rearranged on screen to isolate x.
Covered · The numerical result is shown and checked against the bracketing values before the clip ends.
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