Chapter ISG, Problem 9.8.1.5P

a.

To determine

To find: The type of the function that best models the data in the table and write the reason of selection.

Answer to Problem 9.8.1.5P

Function of linear type should be used.

Explanation of Solution

Given Information: A relevant table.

A linear function best models the data in the table because height is decreasing at uniform rate of $18ft.$ per second.

Thus, function of linear type should be used.

b.

To determine

To find: The rate of change of the function over the interval $x=0tox=4$ and to state its meaning related to motion of the elevator.

Answer to Problem 9.8.1.5P

The rate of change of height is $-18ft./\sec .$ , this implies that elevator is coming down.

Explanation of Solution

Given Information: A relevant table.

Calculation:

  $\frac{124-142}{1-0}=\frac{106-124}{2-1}=\frac{88-106}{3-2}=\frac{70-88}{4-3}=\frac{70-142}{4-0}=-18$

  $\Rightarrow$Rate of change of height is $-18ft./\sec .$

As the height above ground is decreasing over time, elevator is coming down and approaching the ground.

Thus, rate of change of height is $-18ft./\sec .$ , this implies that elevator is coming down.

c.

To determine

To write: A function that models the height of the elevator as a function of time and to explain how the coefficient of $x$ is related to answers in part $aandb.$

Answer to Problem 9.8.1.5P

The function is $y=-18x+142$ , slope is $-18$ which represents the rate of change of height over time.

Explanation of Solution

Given Information: A relevant table.

Calculation:

Slope intercept form: $y=mx+b$

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ \\ m=\frac{124-142}{1-0}\\ \\ =-18\end{array}$

When $x=0$ , $y=142$ ,

  $\Rightarrow \text{y-intercept=142}$

Substituting the values in slope-intercept form, function is,

  $y=-18x+142$

Coefficient of $x$ is $-18$ , it represents slope, as this is constant throughout, linear function was selected in part $a$ . In part $b$ , the rate of change of the function was calculated to be $-18$ which is equal to coefficient of $x$ in the equation of model.

Thus, the function is $y=-18x+142$ , slope is $-18$ which represents the rate of change of height over time.