Question

The tallest escalator on the London Underground system is at angel station. It rises to a height of 90 feet over a horizontal distance of 200 feet. Let e(x) be the height of one of the escalator's steps as a function of the horizontal distance from the bottom step.
The following table shows the coordinates of a linear function.

x ƒ(x)
-5 -1 1/9
-4 -8/9
0 0
2 4/9
3 2/3
Find and compare the rate of change of e(x) and ƒ(x). Which function has the greater rate of change and by how much?

Answer 1

E(x) has the greater rate of change; it is greater by 41/180.

The rate of change of e(x) is given by rise/run:  90/200 = 9/20

The rate of change of f(x) can be found using the formula

m = (y₂ - y₁)/(x₂ - x₁) = (0 - -8/9)/(0 - -4) = 8/9 ÷ 4 = 8/9 × 1/4 = 8/36 = 2/9

Comparing 2/9 and 9/20, we find a common denominator.  180 works:

2/9 = 40/180
9/20 = 81/180

9/20 is the larger slope.

Answer 2

Answer:

The rate of change of e(x) is given by rise/run:  90/200 = 9/20

The rate of change of f(x) can be found using the formula

m = (y₂ - y₁)/(x₂ - x₁) = (0 - -8/9)/(0 - -4) = 8/9 ÷ 4 = 8/9 × 1/4 = 8/36 = 2/9

Comparing 2/9 and 9/20, we find a common denominator.  180 works:

2/9 = 40/180

9/20 = 81/180

9/20 is the larger slope.