# Escalator Motion and Average Rates of Change

Up escalator.

The average rate of change of a function f(x):

the average rate of change of y = f(x) with respect to x over interval [a, b] is given by the quotient

This is illustrated geometrically as shown in Figure 1 and we say

Figure 1.

that the quotient  Dy / Dx is the slope of the secant line from (a, f(a)) to (b, f(b)).

Example of AVERAGE RATES OF CHANGE:

An escalator employs a right triangle so that people can move from a lower floor to an upper floor (or vice versa) as shown in Figure 2. We have labeled the base of the triangle horizontal (denoted H), the altitude of the triangle vertical (denoted V), and the hypotenuse of the triangle people (denoted P, for the distance people travel between floors).

Figure 2.

The following animation provides a reasonable visualization of the motion of an escalator.

The gear mechanisms that are used to move the treads of the escalator are set a certain average speed, denote this as s inches per second. Hence the people move between floors along the hypotenuse of the triangle at the average rate DP = s . Natural questions that arise involve the related rates of change both horizontally and vertically. For instance,

• What is the average rate of change DH of the horizontal distance H?

• What is the average rate of change DV of the vertical    distance V?

Our approach to answer these questions depends upon the information we have available concerning the escalator's "dimensions", that is the lengths of the sides of the right triangle and the value of s. We give several scenarios next.

Scenario #1. We are given s = 10"/sec, H = 400", and V = 300". Find  DH and DV.

To determine DH and DV we need to compute the ratios

Hence an intermediate step to compute "time to travel between floors" is required. At this point we need the relationship between P, H, and V given by the Pythagorean theorem and then the calculation

Compute DH and DV.

Thus we have related the average rates of change along the sides of the right triangle.

Scenario #2. For a certain escalator we have measured H = 600" and V = 240". In addition it has been measured by experiment that the average travel time between floors is t = 50 seconds. Find  DH, DV, and DP

DRH  3/24/2002