Calculus I · Unit 2B · lesson

Mixed Related-Rates Problems

Concept

Learning objectives

Handle models where multiple dimensions have independent rates.

Problems with More Than One Changing Dimension

Explanation

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.

Explanation

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.

Guided walkthrough

A rectangle changing in both directions

A rectangle's length increases at $3$ cm/s while its width decreases at $2$ cm/s. How fast is area changing when length is $10$ cm and width is $7$ cm?

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

A balloon with changing radius and density

If mass $M=\rho V$ , then

\frac{dM}{dt}=\rho\frac{dV}{dt}+V\frac{d\rho}{dt}.

The product rule is itself a related-rates relation whenever both factors depend on time.

Exercise

A sphere's radius shrinks at $0.4$ cm/min. Find $dV/dt$ at $r=5$ .

Exercise

Two cars leave an intersection on perpendicular roads at $30$ and $40$ mph. How fast does separation increase after one hour?

Exercise

A $10$ -foot ladder slides with bottom rate $1.5$ ft/s. Find the top rate when $x=6$ .

Exercise

Water fills an inverted cone with height $9$ and radius $3$ at $5$ cubic units/min. Find $dh/dt$ at $h=6$ .

Exercise

A person walks toward a building while the viewing angle to its roof changes. Create a complete variable diagram and rate equation.

Modeling lab

Gas mass in a changing chamber

A chamber contains gas with mass $M(t)=\rho(t)V(t)$ . At one instant, $\rho=1.2$ kg/m $^3$ , $V=10$ m $^3$ , $\rho'=-0.03$ kg/m $^3$ /min, and $V'=0.5$ m $^3$ /min. Then

M'=\rho'V+\rho V'=(-0.03)(10)+(1.2)(0.5)=0.3\text{ kg/min}.

Expansion lowers density, but the increasing volume contributes more strongly, so total mass is rising in this model.

After the explanation

Reading lens

Freeze the geometry at one instant, but differentiate the relationship while every changing quantity is still a function of time.

Mental model

The picture supplies a constraint; implicit differentiation transmits known rates through that constraint to the unknown rate.

Decision

Draw and label first, write one relationship, differentiate with time, then substitute the snapshot measurements and rates.

Common trap

Substituting numerical dimensions before differentiating and thereby erasing the very change the problem asks about.

Check yourself

Does your final rate have the predicted sign, the correct units, and a magnitude compatible with the diagram?

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