Question

The water level, w, in feet, of a river after a rainstorm is a function of the time, t, in hours, since the storm began. The table below shows the water level readings collected at different times. Hours Since Storm Began (t) Water Level (w) 1 18.7 1.5 19.1 2 19.5 Write a linear function that models the data in the table.w(t)= [blank] −−−−−−Enter your answer as a rule of a function.

Answer 1

`w(t)=0.8(x-1)+18.7`

Step-by-step explanation:

If we plot the points in a graph, we can see that they form a line:

Then, we can use the point-slope form of a line.

The slope-point form of a line, given a point P, is:

`\begin{gathered} P=(x_P,y_P) \\ f\mleft(x\mright)=m\lparen x-x_P)+y_P \end{gathered}`

Where

m is the slope

(x_P, y_P) are the coordinates of a point.

To find the slope, we need two points P and Q:

`\begin{gathered} \begin{cases}P=(x_P,y_P) \\ Q=(x_Q,y_Q)\end{cases} \\ . \\ m=(y_P-y_Q)/(x_P-x_Q) \end{gathered}`

Let's take the points:

P = (1, 18.7)

Q = (2, 19.5)

Then, using the formula for the slope:

`m=(18.7-19.5)/(1-2)=(-0.8)/(-1)=0.8`

Now, using the slope-point form of a line, with m = 0.8 and P = (1, 18.7):

`w(t)=0.8(x-1)+18.7`