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Related Rates

In a related rates problem, we are given the rate of change of one quantity and wish to find the rate of change of another. Such problems can often be solved by implicit differentiation.

Example 1

Related rates problems have the following form.

Given:

(a)    Two quantities which depend on time, say x and y.

(b)    The rate of change of one of them, say dx/dt.

(c)    An equation showing the relationship between x and y.

(Usually this information is given in the form of a verbal description of a physical situation and part of the problem is to express it in the form of an equation.)

The problem: Find the rate of change of y, dy/dt, at a certain time t0. (The time t0 is sometimes specified by giving the value which x, or y, has at that time.)

Example 2: Sliding ladder

Related rates problems can frequently be solved in four steps as we did in the examples.

Step 1

Label all quantities in the problem and draw a picture. If the labels are x, y, and t (time), the remaining steps are as follows:

Step 2

Write an equation for the given rate of change dx/dt. Write another equation for the given relation between x and y.

Step 3

Differentiate both sides of the equation relating x and y with respect to t. We choose the time t as the independent variable. The result is a new equation involving x, y, dx/dt, and dy/dt.

Step 4

Set t = t0 and solve for dy/dt. It may be necessary to find the values of x, y, and dx/dt at t = t 0 first.

The hardest step is usually Step 2, because one has to start with the given verbal description of the problem and set it up as a system of formulas. Sometimes more than two quantities that depend on time are given. Here is an example with three:

Example with three quantities dependent on time.(Example 3)

 

Example 4: Population Growing Rate

We conclude with another example from economics. In this example the independent variable is the quantity x of a commodity. The quantity x which can be sold at price p is called the demand function D(p),

x = D(p).

When a quantity x is sold at price p, the revenue is the product

R = px.

The additional revenue from the sale of an additional unit of the commodity is called the marginal revenue and is given by the derivative

marginal revenue = dR/dx.

Example 5

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Last Update: 2006-11-25