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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 ().

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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:

1. Make a sketch of the problem
2. Identify constant and variable quantities
3. Establish relationship between quantities.
4. Differentiate w.r.t time.
5. 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!

LIVEMath

Now for the normal answer:

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?

Answer

 

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.

Answer

 

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.

Answer

 

Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

• Applications of Differentiation
• 1. Tangents and Normals
• 2. Newton's Method
• 3. Curvilinear Motion
• 4. Related Rates
• 5. Curve Sketching Using Differentiation
• 6. More Curve Sketching Using Differentiation
• 7. Applied Maximum and Minimum Problems
• 8. Radius of Curvature

• Comments, Questions?

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