4.5 Solving Related Rates Problems

Here’s a clear step-by-step method you can use on any related-rates problem (fits the AP CED CHA-3.E ideas): 1) Read the context, label quantities and units. Identify which rate is asked for (e.g., dy/dt) and which rates are given. 2) Write an equation linking the quantities (use geometry: Pythagorean, volume formulas, similar triangles, etc.). This is your constraint. 3) Differentiate both sides with respect to time t (use d/dt, implicit differentiation, and the chain rule). Replace derivatives like d/dt[x] with x′ or dx/dt. 4) Plug in the known values (current sizes and known rates)—always use the instantaneous values, not later/earlier ones. 5) Solve algebraically for the unknown rate and include units. Check sign (positive/negative) to match context. 6) Quick sanity check: does the sign/magnitude make sense physically? Common AP set-ups: ladder (Pythag), shadow (similar triangles), filling/emptying tanks (volume dV/dt), cone/sphere formulas. For guided examples and practice, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and drill problems at (https://library.fiveable.me/practice/ap-calculus).

Related-rates problems don’t have one magic formula—the method is the formula. Steps (and the key derivative rule): 1. Relate the quantities with an equation (e.g., V = 4/3 π r^3, x^2 + y^2 = z^2, similar-triangle ratios). 2. Differentiate both sides with respect to time t using implicit differentiation + the chain rule. That gives a linear relation between rates. In general, if F(x,y,...) = 0 then d/dt[F(x,y,...)] = Fx dx/dt + Fy dy/dt + ... = 0 (where Fx means ∂F/∂x evaluated at the current values). 3. Plug in known values for the variables and known rates, then solve for the unknown rate. Example: V = (4/3)πr^3 ⇒ dV/dt = 4πr^2 · dr/dt. On the AP exam use d/dt notation, implicit differentiation, and geometry (Pythagorean, similar triangles) as needed. For a quick topic review see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and practice problems (https://library.fiveable.me/practice/ap-calculus).

Use implicit differentiation whenever the relationship between the quantities is given implicitly (one equation linking variables) rather than as an explicit y = f(x) or one variable in terms of the other. In related rates you usually have an equation relating two or more changing quantities (Pythagorean relation for a ladder, volume formula for a cone, similar-triangle ratios, etc.). Differentiate that equation with respect to time t using the chain rule (every variable that depends on t gets a d/dt and a corresponding time-derivative like dx/dt, dV/dt). Then solve for the unknown rate. Quick checklist: - If you can write one variable explicitly in terms of t and it’s simpler, you may do that, but most word problems give a relation between quantities ⇒ use implicit differentiation. - Always include units and evaluate at the instant (plug in the current measurements) before solving for the unknown rate. This aligns with the CED: CHA-3.E (use derivative to solve related rates using implicit differentiation and chain rule). For more examples and step-by-step practice, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and more unit resources (https://library.fiveable.me/ap-calculus/unit-4). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

Start with these 5 clear steps you can follow every time: 1) Read the problem and label quantities with variables (with units). Identify which rate is asked for (e.g., dy/dt) and which rates are given (e.g., dx/dt). 2) Find an equation that links the quantities (geometry, volume, similar triangles, Pythagorean, etc.). This is the key: write a relationship between the variables, NOT the rates. 3) Differentiate both sides with respect to time t using implicit differentiation and the chain rule (use d/dt and attach a ˙ or d/dt to every variable). Example: for x^2 + y^2 = r^2, 2x dx/dt + 2y dy/dt = 2r dr/dt. 4) Plug in the current values of the variables and the known rates, then solve algebraically for the unknown rate. Keep units (units/sec, etc.). 5) Check sign and reasonability (is the sign consistent with the context?). Quick example: ladder 13 ft long, foot moving away at 2 ft/s, find rate top slides down when foot is 5 ft from wall. Relation: x^2 + y^2 = 13^2. Differentiate: 2x dx/dt + 2y dy/dt = 0 → dy/dt = −(x/y) dx/dt. Plug x=5, y=12, dx/dt=2 → dy/dt = −(5/12)(2) = −5/6 ft/s. For AP guidance and extra examples, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22). For broader unit review or lots of practice problems, check Unit 4 (https://library.fiveable.me/ap-calculus/unit-4) and the practice bank (https://library.fiveable.me/practice/ap-calculus).

Differentiate every quantity that changes with time—basically any variable in your relation that’s a function of t. Quick checklist you can use on every related-rates problem: - Identify the physical quantities (lengths, radius, volume, angle) and pick one as the “given” or “wanted” rate. - Write an equation relating those quantities (Pythagorean, similar triangles, V = πr²h, etc.). - Explicitly label each variable as a function of time: r(t), h(t), x(t), θ(t). Constants (like π or fixed distances) stay constant. - Differentiate both sides with respect to t using implicit differentiation and the chain rule (d/dt[x(t)] = x′(t) = dx/dt). That gives an equation of rates (e.g., 2r dr/dt · h + πr² dh/dt for V). - Plug in known values and solve for the unknown rate. This is exactly CHA-3.E work on the CED (implicit differentiation, chain rule, interpreting d/dt). For extra practice and step-by-step examples (ladder, shadow, cone), see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and more unit resources (https://library.fiveable.me/ap-calculus/unit-4).

Related rates and optimization both use derivatives, but they ask different questions and use different tools. - Related rates: you’re given how one quantity changes in time (d/dt) and asked how another related quantity changes at that instant. Set up an equation linking the quantities (Pythagorean, volume formula, similar triangles), differentiate implicitly with respect to t (use chain rule), then substitute known values to solve for the required d/dt (CED keywords: d/dt notation, implicit differentiation, instantaneous rate of change). Classic: ladder sliding -> relate dx/dt and dy/dt via x^2 + y^2 = L^2. - Optimization: you want the maximum or minimum value of a quantity (often not explicitly time-based). Express that quantity as a function, take derivative with respect to the variable, set f′(x)=0 to find critical points, then use first/second-derivative tests or endpoints to choose the extremum (CED: applying derivatives to analyze functions). Classic: find dimensions that minimize surface area for fixed volume. On the AP exam, related-rates problems focus on interpreting instantaneous rates and using implicit differentiation (Topic 4.5); optimization appears under analyzing functions (Unit 5) and requires justification of critical points. For more related-rates practice and step-by-step guides, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22), the Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4), and thousands of practice problems (https://library.fiveable.me/practice/ap-calculus).

Pick variables, write the relation, differentiate, plug numbers—that’s the recipe. 1) Define: x(t) = distance from wall to ladder base, y(t) = height of ladder top, L = ladder length (constant). So x^2 + y^2 = L^2 (Pythagorean). 2) Differentiate w.r.t. t (implicit differentiation, chain rule): 2x dx/dt + 2y dy/dt = 0 → x dx/dt + y dy/dt = 0. 3) Solve for the unknown rate: dy/dt = −(x/y) · dx/dt. The negative sign means y is decreasing (top sliding down). Example: L = 5 m, base moving away at dx/dt = 2 m/s, and x = 3 m. Then y = sqrt(25 − 9) = 4 m. So dy/dt = −(3/4)(2) = −1.5 m/s. Interpretation: the top is sliding down at 1.5 m/s at that instant. This uses CHA-3.E (implicit differentiation, d/dt notation, instantaneous rates). For extra practice and AP-style problems, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and the practice bank (https://library.fiveable.me/practice/ap-calculus).

You mess up the chain rule in related-rates because you treat variables like constants instead of functions of time. In related-rates problems you almost always use implicit differentiation with respect to t: differentiate every term, and whenever you differentiate a variable (like x, r, θ, h) write its derivative (dx/dt, dr/dt, dθ/dt, dh/dt). That’s the chain rule in action—e.g., d/dt[ r^2 ] = 2r · dr/dt. Quick checklist to avoid mistakes: - Don’t plug numbers in before differentiating. Differentiate the general equation first. - Label which derivatives are known/unknown and include units (helps catch sign errors). - Use implicit differentiation and treat composite quantities (like V = πr^2h) term-by-term: dV/dt = π(2r dr/dt · h + r^2 dh/dt). - Re-check algebra and signs after you substitute values. Practice the standard ladder, shadow, and volume problems until this pattern is automatic. For a targeted review and practice problems, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and hundreds of AP practice questions (https://library.fiveable.me/practice/ap-calculus).

Start with a quick checklist and then an example—that’s the easiest way to solve balloon problems. Steps 1. Draw and label the balloon. Write a relation linking quantities (for a spherical balloon: V = (4/3)πr^3). Use CED keywords: implicit differentiation, chain rule, d/dt. 2. Differentiate both sides with respect to t (time). For the sphere: dV/dt = 4πr^2 · dr/dt. 3. Plug in the given instantaneous values (dV/dt and r at that moment). Solve for the unknown rate (dr/dt). Keep units (e.g., cm^3/s → cm/s). 4. Interpret the sign and meaning (CHA-3.E: interpret related rates in context). Example: If dV/dt = 50 cm^3/s when r = 5 cm, then dr/dt = (dV/dt) / (4πr^2) = 50 / (4π·25) = 50 / (100π) = 0.159/π ≈ 0.159 cm/s. Notes: Always check whether geometry needs similar triangles or Pythagorean relations (shadow/ladder problems). For more worked examples and AP-style practice, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4). For lots of practice problems, go to (https://library.fiveable.me/practice/ap-calculus).

dy/dt means the instantaneous rate of change of y with respect to time t—how fast the quantity y is changing at that exact moment (units: whatever y’s units per unit time). In related-rates problems you don’t solve for y as a function of t first; you 1) write a relationship linking the quantities (e.g., x^2 + y^2 = 25 or V = πr^2h), 2) differentiate both sides with respect to t (use implicit differentiation + chain rule: d/dt[f(y)] = f′(y)·dy/dt), 3) plug in the current values of the variables and any known rates (dx/dt, dr/dt, dV/dt, etc.), and 4) solve algebraically for dy/dt and include units. Example: if x^2 + y^2 = 25, differentiate → 2x(dx/dt) + 2y(dy/dt) = 0, so dy/dt = −[x(dx/dt)]/y. Interpret the numeric sign/units (e.g., decreasing at 3 cm/s). This is exactly the CHA-3.E related-rates idea tested on the AP exam. For step-by-step examples and practice, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22), the Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4), and extra problems (https://library.fiveable.me/practice/ap-calculus).

Differentiate first, plug in values second. Always write a relationship between the changing quantities (use variables), take d/dt implicitly (use chain rule and d/dt notation), then substitute the specific x, y, L, and rate numbers to solve for the unknown rate. Why: differentiation needs the general relationship so each variable’s rate (like dx/dt, dy/dt) appears. If you plug numbers too early you lose the variable you need to differentiate and can’t get the unknown rate. Quick example: ladder of fixed length L: x^2 + y^2 = L^2. Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0. Now plug x, y, and dx/dt to solve for dy/dt. On the AP, show implicit differentiation and d/dt notation (CHA-3.E.1 keywords: implicit differentiation, chain rule, d/dt). For extra practice, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and more problems (https://library.fiveable.me/practice/ap-calculus).

Start your diagram by thinking: what geometric picture matches the words? Draw that shape clearly and label every variable as a function of time (use x(t), h(t), r(t) or just x, h with d/dt notation). Steps: 1. Sketch the physical setup (ladder, cone, shadow, cylinder, etc.)—include right angles, centers, ground line, water surface. 2. Label lengths/angles you’ll relate (e.g., distance from wall = x, height of top = y, radius = r). Write units next to labels. 3. Mark known rates (dx/dt, dh/dt) and the unknown rate you must find (dy/dt) with arrows showing direction of change. 4. Add geometry relations: Pythagorean, similar triangles, volume formula, etc. Put the equation near the diagram. 5. Circle the instant (time) you evaluate and note values to substitute. 6. When you differentiate implicitly, keep d/dt on every variable. Good diagrams make implicit differentiation and substitution straightforward. For more practice and AP-aligned examples, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22), the Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4), and tons of practice problems (https://library.fiveable.me/practice/ap-calculus).

Most common related-rates setups you'll see on the AP exam: - Ladder sliding (use Pythagorean: x^2 + y^2 = L^2)—find dx/dt or dy/dt. - Shadow/light problems (similar triangles relating object, shadow, and distance). - Expanding/contracting circles and spheres (dA/dt, dV/dt using A = πr^2, V = (4/3)πr^3). - Filling/draining cones or tanks (related r and h via similar triangles; use V for cone). - Cylinders (V = πr^2h with dr/dt or dh/dt nonzero). - Distance between moving objects (use Pythagorean or relative-position functions). - Angle rates (dθ/dt) problems using trig relations or implicit differentiation. - Rates in motion contexts (radial velocity, connecting position/velocity/acceleration). Key moves AP expects: set up an equation relating quantities, differentiate implicitly with d/dt, plug known values (including sign/units), then solve. Topic 4.5 maps to CHA-3.E in the CED (use chain rule, implicit differentiation, similar triangles). For targeted review and practice, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22), the full Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4), and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

Getting a negative rate is normal—it just means that quantity is decreasing (or moving in the opposite direction of the sign convention you picked). In related-rates problems you use d/dt (implicit differentiation) to link rates; the algebra can produce negative values when the physical quantity is shrinking. Quick checklist to avoid sign surprises: - Define each variable and a clear sign convention (e.g., height h positive upward, distance x positive along ground). Draw a picture. - Write the correct relation (Pythagorean, similar triangles, V = …). - Differentiate implicitly with respect to t using d/dt notation and chain rule (CED keywords: implicit differentiation, chain rule). - Plug in values and units. If you get negative, interpret it: e.g., dh/dt = −3 cm/s means height is decreasing 3 cm each second. - If interpretation seems wrong, recheck your sign choices and algebra (especially product/quotient rule, signs when differentiating constants). For more worked examples and practice aligned to Topic 4.5, see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and extra practice problems (https://library.fiveable.me/practice/ap-calculus).

Start with the geometry and write one equation that links the quantities. For a right circular cone with fixed full height H and base radius R, similar triangles give r = (R/H)·h. Use the cone volume formula V = (1/3)πr^2h and replace r: - V = (1/3)π (R^2/H^2) h^3. Differentiate both sides with respect to t (implicit differentiation / chain rule): - dV/dt = π (R^2/H^2) h^2 · dh/dt. Solve for the rate you want. For example, to find dh/dt: - dh/dt = (dV/dt) · (H^2) / (π R^2 h^2). How to use it on an AP problem: 1. Identify which rates are given (usually dV/dt known) and which you want (dh/dt or dr/dt). 2. Plug numeric values for H, R, current h, and dV/dt (keep consistent units). 3. Remember units (e.g., m^3/min for dV/dt gives m/min for dh/dt) and that answers are instantaneous rates. This follows CED skills: similar triangles, implicit differentiation, d/dt notation. For worked examples and more practice, see the Topic 4.5 study guide (https://library.fiveable.me/ap-calculus/unit-4/solving-related-rates-problems/study-guide/oXqNN9mrHM2r16Pjzm22) and extra practice questions (https://library.fiveable.me/practice/ap-calculus).