Related Rates
If two variables are related, their rates of change are related. These are real-world word problems powered by implicit differentiation.
The Concept
Differentiation with Respect to Time
In these problems, every variable () is implicitly a function of time .
- Derivative of is
- Derivative of is
General Strategy
- Draw a picture and label variables.
- Find an equation relating the variables.
- Differentiate both sides with respect to .
- Plug in known values and solve.
Worked Example
Inflating Balloon
A spherical balloon is inflating. The radius grows at . How fast is the Volume () changing when ?
Volume of a sphere:
Remember the Chain Rule for r!
Use and .
Level Up Examples
The Sliding Ladder
A 15 ft ladder rests against a wall. The bottom is initially 10 ft away and is pushed towards the wall at . How fast is the top moving up the wall after 12 seconds?
Start at . Rate .
. (Set slider to 7!)
If , then .
The Streetlight Shadow
A 12 ft pole. A 5.5 ft person walks away at 2 ft/s. Find the rate of the shadow's tip when they are 25 ft away.
Let be the tip position. (Big triangle vs Small triangle)
Since is a linear relationship, the rates are proportional constant!
Given . Note: Distance doesn't change the rate!
Practice Quiz
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