4. Related Rates
by M. Bourne
If 2 variables both vary with respect to time and have a relation between them, we can express the rate of change of one in terms of the other.
We need to differentiate both sides w.r.t.
(with respect to) time (
).
Important!
Recall from implicit differentiation the following:
We use these throughout this section on related rates.
Example 1:
A 20 m ladder leans against a wall. The top slides down at a rate of 4 ms-1. How fast is the bottom of the ladder moving when it is 16 m from the wall?
Steps:
- Make a sketch of the problem
- Identify constant and variable quantities
- Establish relationship between quantities.
- Differentiate w.r.t time.
- Evaluate at point of interest.
Following is a demonstration of how to solve this problem in LiveMath.
You obviously still need to understand the mathematics!
Now for the normal answer:
Example 2:
A stone is dropped into a pond, the ripples forming concentric circles which expand. At what rate is the area of one of these circles increasing when the radius is 4 m and increasing at the rate of 0.5 ms-1?
Example 3:
An earth satellite moves in a path that can be described by
where x and y are in thousands of kilometres.
If dx/dt = 12900 km/h for x = 3200 km and y > 0, find dy/dt.
Example 4:
The tuning frequency f of an electronic tuner is inversely proportional to the square root of the capacitance C in the circuit.
If f = 920 kHz for C = 3.5 pF, find how fast f is changing at this frequency if dC/dt = 0.3 pF/s.
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