Suppose an elevator is 400 feet above the ground. It descends at a steady rate. After 15 seconds it is 250 feet above the ground.

Question

a. Write a linear function for height of the elevator as a function of time

b. Is it reasonable to include negative numbers in a range?

Answer 1

Let's start with some concept review, since the questions both involve a conceptual understanding of linear functions.

So, part (a) references height as a function of time. That means that in this case, the height is our dependent variable; it literally varies depending on what time we choose. Time is the independent variable – we can select a time at random and the function will tell us where the elevator is at that time. So, if we were to graph this, the time would be our x and the height would be our y.

So we need to write a linear function for the scenario given, with time as the x and height as the y. Let's start with the most-common form for a linear equation: slope-intercept form:

y = mx + b

Where m is the slope and b is the y-intercept. Our next order of business is to figure out what those two constants are, since the problem doesn't give them to us.

The problem does helpfully give us two points to work with – one at the beginning and one 15 seconds in. The key here is the assumption that the starting value of 400 feet corresponds to Time = 0 – knowing that, we can write our two points as:

(0, 400)

(15, 250)

We only need two points to determine the slope, so let's start there:

m = (y2 – y1)/(x2 – x1)

m = (250 – 400)/(15 – 0)

m = -150/15

m = -10

So our slope is negative 10 – which makes sense, since the line is descending and should have a negative slope.

Y = -10x + b

Now about that y-intercept – how do we figure that out?

Well, the y-intercept is the place where the line crosses the y-axis – in other words, the value of y when x is 0. Well, we just so happen to have a point where x is 0 already in our givens! So 400 is our y-intercept!

Y = -10x + 400

And that's the answer to part (a).

Now, on to part (b). Is it reasonable to include negative numbers in the range?

Well, remember, the 'range' is another term for the set of all possible y values. (The 'domain' is the term for the set of possible x values.) The y-axis in our scenario depicts the height of the elevator off the ground, so theoretically y = 0 means ground level. But since we're talking about an elevator, and elevators frequently go below ground level, I don't see any reason why we couldn't include negative numbers in the range, unless the problem specifies that the elevator doesn't go to the basement!

In contrast, it doesn't make sense to include negative numbers in our domain, because we're assuming time to start at x = 0. Negative time doesn't make much sense unless there's a specific reason to be counting down to a time 0 (maybe a space shuttle launch?). All things being equal, it'd make sense to include negative numbers in the range but not the domain.