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.
Finding Linear Functions
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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.
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
(0 seconds, 400 feet)
(15 seconds, 250 feet)
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...