Calculus I · 2B · lesson
Mixed Related-Rates Problems
Learn mixed related-rates problems with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Related ratesWhat this section is building
Learn mixed related-rates problems with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
The picture supplies a constraint; implicit differentiation transmits known rates through that constraint to the unknown rate.
Draw and label first, write one relationship, differentiate with time, then substitute the snapshot measurements and rates.
Substituting numerical dimensions before differentiating and thereby erasing the very change the problem asks about.
Learning objectives
Handle models where multiple dimensions have independent rates.
Problems with More Than One Changing Dimension
Before the formulas
Related-rates work in Mixed Related-Rates Problems becomes manageable when geometry and calculus are kept separate. Geometry supplies an equation such as the Pythagorean theorem, a volume formula, or a similar-triangle proportion. Calculus differentiates that equation. The derivative does not invent the geometry for you.
Before solving, predict the sign of the unknown rate from the picture. That prediction is a powerful error check. If a ladder foot moves away from a wall, the top should move down; a positive answer for its height rate should trigger a review.
Mixed problems still use the same five-step skeleton
Name variables, draw the relationship, differentiate with respect to time, insert the instant, and solve for the requested rate. The surface story may involve water, aircraft, cameras, or expanding objects, but the mathematical skeleton remains stable.
After solving, interpret the sign. A negative rate is often meaningful rather than wrong: a height may be falling, a distance shrinking, or an angle closing.
Some systems contain several independently changing dimensions. The product rule naturally appears in changing areas, masses, densities, and energies because more than one factor contributes to the total rate.
A negative term does not automatically make the final rate negative. Competing effects can reinforce or cancel. The arithmetic at the end should be interpreted as a balance of mechanisms, not merely a number produced by a formula.
A rectangle changing in both directions
A rectangle's length increases at cm/s while its width decreases at cm/s. How fast is area changing when length is cm and width is cm?
Worked solution
Write a real attempt before opening the supplied answer.
A balloon with changing radius and density
If mass , then
The product rule is itself a related-rates relation whenever both factors depend on time.
A sphere's radius shrinks at cm/min. Find at .
Two cars leave an intersection on perpendicular roads at and mph. How fast does separation increase after one hour?
A -foot ladder slides with bottom rate ft/s. Find the top rate when .
Water fills an inverted cone with height and radius at cubic units/min. Find at .
A person walks toward a building while the viewing angle to its roof changes. Create a complete variable diagram and rate equation.
Gas mass in a changing chamber
A chamber contains gas with mass . At one instant, kg/m, m, kg/m/min, and m/min. Then
Expansion lowers density, but the increasing volume contributes more strongly, so total mass is rising in this model.
Source & rights
Original instruction with traceable references.
BetterGrades-original composition declared by source handoff; owner provenance review required before public release
Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.