Premed · Premed · Calculus 1
Lecture 10: Related Rates
Calculus I — Differential Calculus
Learning Objectives
By the end of this lecture, students will be able to:
- Identify related rates problems and set up the appropriate equations
- Apply implicit differentiation with respect to time
- Follow a systematic strategy for solving related rates problems
- Solve classic related rates problems (expanding shapes, ladder problems, shadow problems)
- Interpret the sign and magnitude of rates in context
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 = 2pir (dr/dt). Substituting dr/dt = 2 m/s and r = 5 m yields dA/dt = 2pi(5)(2) = 20pi 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.
<image>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."</image>
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)pir^2h = (1/3)pi(h/2)^2h = pih^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/(4pi) m/min, approximately 0.239 m/min.
<image>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."</image>
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.
<image>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."</image>
