Solving Related Rates Problems
Build a geometric relationship, differentiate before inserting a snapshot, and interpret the resulting signed rate.
College Board Topic 4.5 · CED p. 86 (PDF p. 91) · CED effective Fall 2020; project course outline 2026–27.
A reliable related-rates solution reads like a small story: draw or describe the geometry, name the changing variables, write the relationship, differentiate using the shared independent variable, substitute the instant’s data, and interpret the signed answer. Include units at the end and check whether the sign fits the picture.
For a 5-meter ladder, x is the foot’s horizontal distance from the wall and y is its top’s height. The equation is x² + y² = 25. When x = 3, y = 4. If x′ = 0.2, then 2xx′ + 2yy′ = 0 gives y′ = −(3/4)(0.2) = −0.15 m/s. The top is sliding downward at 0.15 m/s. The negative sign belongs to the upward-positive height variable.
Sometimes geometry supplies a missing relation. In a conical tank, similar triangles can express radius in terms of water depth. Those are the dimensions of the water, not necessarily the dimensions of the whole tank. A shadow problem often needs a similar-triangle relation using the full tip distance, while a distance problem needs the Pythagorean theorem. Choose a sensible coordinate system rather than guessing a familiar formula.
Numerical tools help with awkward values, but cannot decide the variables, constraints, or signs. At configurations where your rate equation would divide by zero, pause: the geometry may make the requested rate undefined or demand a separate analysis. Never conceal this by reporting a calculator error.
Let's work through it
A conical tank has total height 6 m and top radius 3 m. Similar triangles give r/h = 3/6, so r = h/2.
Why: The water cone stays similar to the tank; this relation holds for every water depth.
V = (1/3)πr²h = πh³/12, hence V′ = (πh²/4)h′.
Why: Eliminate the changing radius before differentiating; then use the chain rule.
When h = 2 m and water enters at V′ = 3 m³/min, h′ = 3/π m/min.
Why: The differentiated relation connects the known volume rate to the unknown depth rate.
The water depth is rising at 3/π m/min at that instant.
Why: The positive value and length/time units match the physical question.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
A distance variable measures separation. What does a negative distance rate mean?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
A distance variable measures separation. What does a negative distance rate mean?
Worked answer and reason
The objects are getting closer at that instant.
The sign describes change in the defined separation.
Retrieval 2
In a cone, why not use the tank’s top radius for shallow water?
Worked answer and reason
The water surface radius changes with depth and is smaller than the top radius.
Similar triangles relate the water dimensions at every instant.
Catch the tempting mistake
- A negative related rate is physically impossible.
- It usually indicates decrease in a chosen positive quantity, such as height or separation.
- The shadow length and the shadow tip’s distance from the lamp are the same.
- The tip distance includes both the person’s distance and the shadow length.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 4 problem 16 · No calculator · analytic
- Unit 4 problem 17 · No calculator · multiple
- Unit 4 problem 18 · No calculator · analytic
- Unit 4 problem 19 · No calculator · multiple
- Unit 4 problem 20 · No calculator · multiple
- Unit 4 problem 21 · No calculator · multiple
For the teacher
- Have groups critique the variable definitions and geometry before seeing any derivative.
- Award separate formative credit for relationship, differentiation, substitution, and contextual sentence; this locates the source of an error.