Chapter 1.4, Problem 44E

To determine

To calculate: The periods during which the surface of elevation is changing linearly.

$\begin{array}{cc}\text{Year, Y}& \text{Elevation, E (ft)}\\ 1965& 4193\\ 1970& 4196\\ 1975& 4199\\ 1980& 4199\\ 1985& 4206\\ 1990& 4203\\ 1995& 4200\end{array}$

Answer to Problem 44E

During the period 1965 to 1975 and 1985 to 1995, the surface of elevation is changing linearly.

Explanation of Solution

Given information:

The table that provides elevation of the surface of Great Salt Lake in Utah for different years.

  $\begin{array}{cc}\text{Year, Y}& \text{Elevation, E (ft)}\\ 1965& 4193\\ 1970& 4196\\ 1975& 4199\\ 1980& 4199\\ 1985& 4206\\ 1990& 4203\\ 1995& 4200\end{array}$

Formula used:

The slope is $m=\frac{y-y_0}{x-x_0}$ where $\left(x,y\right)$ is an arbitrary point and $\left(x_0,y_0\right)$ is the base point.

Calculation:

Consider the table that provides elevation of the surface of Great Salt Lake in Utah for different years.

  $\begin{array}{cc}\text{Year, Y}& \text{Elevation, E (ft)}\\ 1965& 4193\\ 1970& 4196\\ 1975& 4199\\ 1980& 4199\\ 1985& 4206\\ 1990& 4203\\ 1995& 4200\end{array}$

The horizontal axis that is x- axis, denote the year and the vertical axis that is y -axis denote the elevation.

Recall slope is $m=\frac{y-y_0}{x-x_0}$ where $\left(x,y\right)$ is an arbitrary point and $\left(x_0,y_0\right)$ is the base point.

Slope between 1965 and 1970 is the slope between the points $\left(1965,4193\right)$ and $\left(1970,4196\right)$ is, here arbitrary point is $\left(1970,4196\right)$ and base point is $\left(1965,4193\right)$ .

Slope of the line is,

$\begin{array}{c}m=\frac{y-y_0}{x-x_0}\\ =\frac{4196-4193}{1970-1965}\\ =\frac{3}{5}\end{array}$

Therefore, slope is $3\text{ ft}$ every 5 years.

Slope between 1970 and 1975 is the slope between the points $\left(1970,4196\right)$ and $\left(1975,4199\right)$ is, here arbitrary point is $\left(1975,4199\right)$ and base point is $\left(1970,4196\right)$ .

Slope of the line is,

$\begin{array}{c}m=\frac{y-y_0}{x-x_0}\\ =\frac{4199-4196}{1975-1970}\\ =\frac{3}{5}\end{array}$

Therefore, slope is $3\text{ ft}$ every 5 years.

Slope between 1975 and 1980 is the slope between the points $\left(1975,4199\right)$ and $\left(1980,4199\right)$ is, here arbitrary point is $\left(1980,4199\right)$ and base point is $\left(1975,4199\right)$ .

Slope of the line is,

$\begin{array}{c}m=\frac{y-y_0}{x-x_0}\\ =\frac{4199-4199}{1980-1975}\\ =\frac{0}{5}\\ =0\end{array}$

Therefore, slope is 0. No change in surface of elevation.

Slope between 1980 and 1985 is the slope between the points $\left(1980,4199\right)$ and $\left(1985,4206\right)$ is, here arbitrary point is $\left(1985,4206\right)$ and base point is $\left(1980,4199\right)$ .

Slope of the line is,

$\begin{array}{c}m=\frac{y-y_0}{x-x_0}\\ =\frac{4206-4199}{1985-1980}\\ =\frac{7}{5}\end{array}$

Therefore, slope is $7\text{ ft}$ every 5 years.

Slope between 1980 and 1985 is the slope between the points $\left(1980,4199\right)$ and $\left(1985,4206\right)$ is, here arbitrary point is $\left(1985,4206\right)$ and base point is $\left(1980,4199\right)$ .

Slope of the line is,

$\begin{array}{c}m=\frac{y-y_0}{x-x_0}\\ =\frac{4206-4199}{1985-1980}\\ =\frac{7}{5}\end{array}$

Therefore, slope is $7\text{ ft}$ every 5 years.

Slope between 1985 and 1990 is the slope between the points $\left(1985,4206\right)$ and $\left(1990,4203\right)$ is, here arbitrary point is $\left(1990,4203\right)$ and base point is $\left(1985,4206\right)$ .

Slope of the line is,

$\begin{array}{c}m=\frac{y-y_0}{x-x_0}\\ =\frac{4203-4206}{1990-1985}\\ =\frac{-3}{5}\end{array}$

Therefore, slope is decreasing by $3\text{ ft}$ every 5 years.

Slope between 1990 and 1995 is the slope between the points $\left(1995,4200\right)$ and $\left(1990,4203\right)$ is, here arbitrary point is $\left(1995,4200\right)$ and base point is $\left(1990,4203\right)$ .

Slope of the line is,

$\begin{array}{c}m=\frac{y-y_0}{x-x_0}\\ =\frac{4200-4203}{1995-1990}\\ =\frac{-3}{5}\end{array}$

Therefore, slope is decreasing by $3\text{ ft}$ every 5 years.

Years during which slope is same depicts that surface of elevation is changing linearly.

Thus, during the period 1965 to 1975 and 1985 to 1995, the surface of elevation is changing linearly.