Related Rates
Related Rates
When quantities change together
Related rates problems ask: If one quantity is changing at a certain rate, how fast is a related quantity changing?
Example: A ladder is sliding down a wall. If the bottom moves at 2 ft/sec, how fast is the top falling?
Classic Related Rates: The Sliding Ladder
As the base slides out, how fast does the top slide down?
Strategy for Related Rates
1. Draw a picture and label variables
2. Write an equation relating the quantities
3. Differentiate both sides with respect to time (t)
4. Substitute known values
5. Solve for the unknown rate
Classic Ladder Problem
Hard
A 10-ft ladder leans against a wall. The bottom slides away at 2 ft/s. How fast is the top falling when the bottom is 6 ft from the wall?
1
Draw and label
Let x = distance from wall to bottom, y = height on wall
Ladder length = 10 ft
2
Write equation
Pythagorean theorem: x^2 + y^2 = 100
3
Differentiate w.r.t. time
2x(dx/dt) + 2y(dy/dt) = 0
4
Find y when x = 6
36 + y^2 = 100, so y = 8
5
Substitute and solve
2(6)(2) + 2(8)(dy/dt) = 0
24 + 16(dy/dt) = 0
dy/dt = -3/2 ft/s
Answer:
The top is falling at 1.5 ft/s
💡 The negative sign means y is decreasing (falling)
Practice
1. A circle's radius grows at 3 cm/s. How fast is the area growing when r = 5 cm?
A = πr². dA/dt = 2πr(dr/dt) = 2π(5)(3) = 30π cm²/s
Key Takeaways
Related rates: find how fast one quantity changes given another's rate
Draw a picture, write equation, differentiate with respect to time
Don't forget the chain rule when differentiating!
Substitute values AFTER differentiating