Ladder and shadow problems are classic related-rates examples because they turn moving geometry into calculus. A ladder sliding down a wall forms a right triangle whose side lengths change with time. A person walking near a streetlight creates similar triangles as the shadow length changes.
These situations matter because they show how derivatives connect quantities that are changing together, even when only one rate is directly given.
The main strategy is to write an equation that relates the changing variables, then differentiate the whole equation with respect to time. For the ladder, the Pythagorean theorem usually gives x^2 + y^2 = L^2, where L is constant. For the shadow, similar triangles connect the streetlight height, person height, distance from the light, and shadow length.
After differentiating, substitute the known measurements and rates to solve for the unknown rate.
Key Facts
- Related rates use derivatives with respect to time, such as dx/dt, dy/dt, and ds/dt.
- For a ladder of constant length L, x^2 + y^2 = L^2.
- Differentiating x^2 + y^2 = L^2 gives 2x dx/dt + 2y dy/dt = 0.
- In ladder problems, if the base moves away from the wall, dx/dt is positive and dy/dt is usually negative.
- For a streetlight shadow, similar triangles often give H/(x + s) = h/s, where H is light height, h is person height, x is distance from light, and s is shadow length.
- If H/(x + s) = h/s, then (H - h)s = hx and ds/dt = h dx/dt/(H - h).
Vocabulary
- Related rates
- A calculus method for finding how fast one quantity changes by using an equation that relates it to other changing quantities.
- Derivative with respect to time
- A derivative such as dx/dt that measures how quickly a variable changes as time changes.
- Pythagorean theorem
- The right-triangle relationship x^2 + y^2 = L^2, often used in sliding ladder problems.
- Similar triangles
- Triangles with equal corresponding angles whose side lengths are proportional.
- Chain rule
- A differentiation rule that accounts for variables changing with time, such as d(x^2)/dt = 2x dx/dt.
Common Mistakes to Avoid
- Forgetting to differentiate with respect to time is wrong because x and y are changing variables, so d(x^2)/dt must be 2x dx/dt, not just 2x.
- Substituting numbers before differentiating can be wrong because it may turn changing variables into constants and erase the rates you need to find.
- Using the wrong sign for a rate is wrong because direction matters. In a ladder problem, the base moving away from the wall has dx/dt > 0 while the top moving down has dy/dt < 0.
- Mixing up the person's distance and the shadow length is wrong because the full distance from the streetlight to the shadow tip is x + s, not just s.
Practice Questions
- 1 A 10 m ladder leans against a wall. The base is 6 m from the wall and moves away at 0.5 m/s. How fast is the top of the ladder sliding down at that instant?
- 2 A 1.8 m tall person walks away from a 6 m streetlight at 1.2 m/s. How fast is the length of the person's shadow increasing?
- 3 In a related-rates problem, explain why the equation should be differentiated before substituting the instant-specific values for the changing variables.