[ d(t) = r t ] 1. Consider 32 feet for the total distance from Point B to Point E on the escalator diagram below. Suppose this escalator is running at a speed of 100 feet per minute.
1-a. Show that 100 feet per minute is equal to 20 inches per second.
1-b. Find a value in miles per hour corresponding to 20 inches per second.
[ d(t) = (5 feet/3 second) t ] 2. Let rate r be 5/3 feet per second, then the distance, d, in feet traveled along the 32 feet of escalator length depends on the product of 5/3 times and the amount of time in seconds, t, starting from Point B. Calculate d(10) = ____________ feet where 10 is the number of seconds spent on the escalator. Substitution of 10 for t in
is indicated by the d of ten, d(10), notation.
( 10, d(10) ) is an ordered pair of numbers you may plot on graph paper where the horizontal axis is time in seconds and the vertical axis is the distance traveled.
[ d(t) = (5 feet/3 second) t ] 3-a. Find d(0) = ____________ feet and (0, ______).
[ d(t) = (5 feet/3 second) t ] 4-a. Graph each point found in problem number 3. Then draw a line from your graphed point to graphed point.
4-b. Write a few sentences about your most interesting ideas that came to you about your graphing.
[ d(t) = (5 feet/3 second) t ] 5-a. Find
5-b. Does the answer to problem number 5-a surprise you? Write your reactions.
[ d(t) = (5 feet/3 second) t ] 6. We can replace the ratio for two exact points:
with one of the points, (12, d(12)), fixed and the other, (t, d(t)), changing and have the same results found in problem number 5.
For example, let t= 12 + h then find an answer for:
1 and 2. For problems 1 and 2 remember 60 seconds per minute, 60 minutes per hour, 12 inches per foot and 5280 feet per mile. It helps to remember your typical walking speed, approximate speeds for cars on city streets or highways, and other faster examples like airplanes all moving at a constant or unchanging speed for a short time. Thinking of these examples will help you determine estimations to consider how accurate your answers are.
3. The relation y = m x + b, often used in math classes such as algebra, is sometimes written f(x) = m x + b to indicate the function idea and to show the input x number and the output y answer for each calculation just as we are suggesting in problem number 3.
4. When you draw a straight line from one point to the next you do not first consider that all of these short line segments will be one longer line segment or that all the points belong to the same line segment. This fact that all points are on the same line for (d(t) = r t is one reason we call (d(t) = r t a linear function.
5-a. The numerator (d(14) - d(12)) may catch you off guard as to what you need to do first. So first find a number replacement for d(14) then a number replacement for d(12). With these substitutions (d(14) - d(12)) is one number subtracted from another. You are right, the denominator (14 - 12) is equal to 2 so now you may finish this division.
6. Realization that you may substitute a variable expression such as twelve plus h for the variable t in (d(t) = r t is the next step forward in understanding how useful function notation may be. So now consider using
in the numerator and (12 + h) in the denominator to get: