# Lecture 10: Related Rates
## Calculus I — Differential Calculus
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## Learning Objectives
By the end of this lecture, students will be able to:
1. Identify related rates problems and set up the appropriate equations
2. Apply implicit differentiation with respect to time
3. Follow a systematic strategy for solving related rates problems
4. Solve classic related rates problems (expanding shapes, ladder problems, shadow problems)
5. Interpret the sign and magnitude of rates in context
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## Lecture Content
### I. What Are Related Rates?
In many real-world scenarios, several quantities change simultaneously, and their rates of change are linked through an equation. **Related rates problems** ask: given the rate of change of one quantity, find the rate of change of another. The key idea is to differentiate a known relationship between variables **with respect to time t**, using the chain rule. All quantities are treated as functions of time, even when the time dependence is not written explicitly.
### II. Strategy for Solving Related Rates Problems
A systematic six-step strategy makes related rates problems manageable. **Step 1**: Draw a diagram and label all variables (quantities that change with time). **Step 2**: Identify what is given (known rates such as dx/dt = 3 m/s) and what is asked (the unknown rate, such as dy/dt when x = 5). **Step 3**: Write an equation relating the variables, using geometric formulas (Pythagorean theorem, area, volume, similar triangles, trigonometry) or physical laws. Crucially, do not substitute specific numerical values until after differentiating. **Step 4**: Differentiate both sides with respect to t using implicit differentiation and the chain rule. **Step 5**: Substitute the known values and solve for the unknown rate. **Step 6**: Interpret the answer, including units and the sign (positive for increasing, negative for decreasing).
### III. Example — Expanding Circle
A stone is dropped into a pond, creating a circular ripple whose radius increases at 2 m/s. The question is: how fast is the area increasing when the radius is 5 m? The relevant variables are r(t) for the radius and A(t) for the area, connected by the equation A = pi * r^2. Differentiating with respect to time gives dA/dt = 2*pi*r * (dr/dt). Substituting dr/dt = 2 m/s and r = 5 m yields dA/dt = 2*pi*(5)*(2) = 20*pi m^2/s, which is approximately 62.8 m^2/s.
### IV. Example — The Sliding Ladder
A 10-foot ladder leans against a vertical wall, and the bottom slides away from the wall at 1 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall? Let x denote the distance from the base of the wall to the bottom of the ladder and y the height of the top. The Pythagorean theorem gives x^2 + y^2 = 100 (the ladder length is constant). Differentiating yields 2x(dx/dt) + 2y(dy/dt) = 0. When x = 6, we find y = sqrt(100 - 36) = 8. Substituting gives 2(6)(1) + 2(8)(dy/dt) = 0, so dy/dt = -12/16 = -3/4 ft/s. The negative sign confirms that the top is sliding **down**.
A diagram of the sliding ladder problem. A vertical wall on the left, a horizontal floor along the bottom. A ladder of length 10 ft connects a point on the wall (height y) to a point on the floor (distance x from the wall). The right triangle is clearly labeled with x, y, and hypotenuse 10. Arrows show dx/dt pointing rightward (positive, base moving away from wall) and dy/dt pointing downward (negative, top sliding down). The equation x^2 + y^2 = 100 is displayed. Title: "The Sliding Ladder Problem."
### V. Example — Filling a Conical Tank
Water is poured into a cone-shaped tank (vertex down) at a rate of 3 m^3/min. The tank has height 10 m and radius 5 m at the top. How fast is the water level rising when the depth is 4 m? The variables are V for volume, h for depth, and r for the radius of the water surface. By similar triangles, r/h = 5/10 = 1/2, so r = h/2. The volume formula becomes V = (1/3)*pi*r^2*h = (1/3)*pi*(h/2)^2*h = pi*h^3/12. Differentiating gives dV/dt = (pi/4)*h^2*(dh/dt). Substituting dV/dt = 3 and h = 4 yields 3 = (pi/4)*(16)*(dh/dt), so dh/dt = 3/(4*pi) m/min, approximately 0.239 m/min.
A cross-section of the conical tank problem. The cone is shown with vertex at the bottom, opening upward. The total height is 10 m and top radius is 5 m. Water fills the cone to a depth h, with the water surface having radius r. A dashed line shows the similar triangles relationship r/h = 5/10. The incoming water flow dV/dt = 3 m^3/min is indicated with an arrow at the top. Labels: h, r, and the similar triangle proportion. Title: "Filling a Conical Tank."
### VI. Example — Shadow Problem
A 6-ft tall person walks away from a 15-ft streetlight at 4 ft/s. How fast is the tip of the shadow moving? Let x denote the distance from the person to the light pole and s the length of the shadow. By similar triangles, 15/(x + s) = 6/s, which simplifies to 15s = 6x + 6s, giving 9s = 6x, so s = (2/3)x. The tip of the shadow is at distance x + s = x + (2/3)x = (5/3)x from the pole. Differentiating gives d/dt(x + s) = (5/3)(dx/dt) = (5/3)(4) = 20/3 ft/s, approximately 6.67 ft/s.
### VII. Common Pitfalls
Several mistakes arise frequently in related rates problems. **Substituting values too early** is the most common: plugging in numerical values before differentiating loses the derivative terms entirely. **Forgetting the chain rule** is another hazard: every variable that depends on t must be differentiated accordingly, so d/dt[r^2] = 2r * dr/dt, not simply 2r. **Sign errors** require attention to whether quantities are increasing (positive rate) or decreasing (negative rate). **Units** should always be included in the final answer, and dimensional consistency should be verified. Finally, **not identifying the correct equation** can derail a solution from the start, so sketching the geometry carefully and distinguishing what stays constant from what changes is essential.
A common-mistakes checklist formatted as a visual guide. Four warning boxes: (1) "Do NOT substitute numbers before differentiating" with a crossed-out incorrect example. (2) "Do NOT forget dy/dt when differentiating y — chain rule!" with a correct vs. incorrect comparison. (3) "Check the sign — positive means increasing, negative means decreasing." (4) "Always include units in your final answer." Title: "Related Rates: Common Mistakes to Avoid."