Finding Linear Functions

Question

The problem is "Suppose an elevator is 400 feet above the ground. It descends at a steady rate. After 15 seconds, the elevator is 250 ft above the ground." I need to find a linear function for the height of the elevator as a function of time. I know it is descending 150 ft per 15 seconds, but I don't know what to do from there.

Answer 1

The general linear equation is of the form y = mx + b

We are required to form a linear equation for the elevation of the elevator from the ground in terms of time.  If we call y as the height and x as the time, then applying the general linear model y = mx + b, the question reduces to finding m and b using the information given in the question.  At time x = 0 seconds the height y is 400 feet.  Applying this data in our model, we get 400 = m * 0 + b

 

The above equation reduces to b = 400

 

Using the value of b above in our model, it becomes y = mx + 400

 

When x = 15 seconds we know that the height is 250.  Applying this data in the model equation above, we get 

 

250 = 15 * m + 400

 

or m = -10

 

So our linear model is y = -10x + 400 where x is time in seconds and y is the elevation of the elevator with reference to the ground expressed in feet.

Answer 2

Hi Ben;

The elevator "descends at a steady rate."  This means that when we place all the points on a graph, it will be a straight line.  You must proceed as if it is any other line.

 

y-axis=height of elevator

x-axis=time

 

(0 seconds, 400 feet)

(15 seconds, 250 feet)

 

y=mx+b

y=mx+400

 

b is the value of y when x=0.  It is the y-intercept.

 

Let's establish slope...

As you said, it is descending 150 feet per 15 seconds...

slope=m=(y-y₁)/(x-x₁)

m=(400-250)/(15-0)

m=150/15=10

 

y=10x+400