A storage room begins a day with 20 items, receives 7 and dispatches 9. Its closing count should be 18 if those are the only changes. A measured count of 17 is not fixed by choosing a more elaborate curve: first investigate the missing item or an inaccurate record.
The basic balance is final stock minus initial stock equals inflow minus outflow plus internal production minus internal consumption. Every term must refer to the same system boundary, time interval and quantity.
Predict
Two connected tanks exchange liquid. If the entire pair is the system, should their internal exchange appear in the total balance? Predict the cancellation before writing separate equations. Then change the boundary to include only one tank and explain why the same transfer now matters.
Experiment
Let x and y be amounts in two compartments. A transfer J moves from x to y, external input I enters x and external loss L leaves y.
Compartment
Rate equation
First
x′=I−J
Second
y′=J−L
Combined
(x+y)′=I−L
With I=3,J=5,L=2 in compatible amount-per-time units, x decreases at rate 2, y increases at rate 3 and the total increases at rate 1. Verify each statement. A shrinking first compartment does not imply that the whole system is losing material.
Understand
For a well-mixed tank of constant volume V, let M be solute mass. Liquid enters at rate q with concentration c and leaves at the same volume rate. Inflow solute rate is qc; outflow concentration is M/V, so outflow mass rate is qM/V. Therefore M′=qc−qM/V. The equation follows from a balance plus the constitutive assumption of perfect mixing.
With V=30 liters, q=3 liters per minute and c=2 grams per liter, the equation is M′=6−M/10 in grams per minute. Its equilibrium mass is 60 grams. The derivative is positive below that mass and negative above it. The relaxation time is 10 minutes, which can be read from the removal coefficient after checking its units.
If the volume changes, V must become a state: V′ equals volume inflow minus outflow. Keeping a constant denominator while using unequal flows would silently combine incompatible assumptions. If mixing is incomplete, a single concentration may not represent the outflow, requiring another compartment or measurement model.
Physical positivity also constrains the interpretation. A numerical update that produces negative stock may reflect a time step that is too large, or a discrete demand model that ignores availability. Decide whether unmet demand is lost, deferred or recorded as a separate backorder state. Clipping a negative value to zero without accounting for the missing flow changes the balance.
Apply
Draw a boundary around a small resource system and list every crossing flow. Calculate a balance residual from a four-row ledger containing initial stock, inputs, outputs and final stock. Explain at least two possible causes of a nonzero residual and propose an observation that distinguishes them. Keep measurement error separate from arithmetic error.
Continue
Dimensional analysis next checks whether the proposed terms can even be added. Conservation organizes the equation; units help reject combinations that cannot represent the same quantity.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A stock starts at 12 units, receives 5 and releases 4 during one step. What is the final stock?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A well-mixed tank has volume 20 L, solute mass 6 g and outflow 2 L/min. What is the solute outflow rate in g/min?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Derive a constant-volume well-mixed tank equation with inflow q at concentration c and equal outflow.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show why a transfer between two compartments cancels from their total balance.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
An inventory update predicts −3 items. Diagnose what must change.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Create a balance audit for your investigation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
