Chapter 1.1, Problem 78E

Temperature Variation The bar graph shows the daily high temperatures for Omak, Washington, and Geneseo, New York, during a certain week in June. Let T_O represent the temperature in Omak and T_G the temperature in Geneseo. Calculate T_O − T_G and |T_O − T_G| for each day shown. Which of these two values gives more information?

To determine

To evaluate: The values of the quantities $T_0-T_{\text{G}}$, $|T_{\text{0}}-T_{\text{G}}|$ for each day and which of the values gives more information.

Answer to Problem 78E

The values of the quantities $T_0-T_{\text{G}}$ and $|T_{\text{0}}-T_{\text{G}}|$ for each day is as follows:

	$T_0$	$T_{\text{G}}$	$T_0-T_{\text{G}}$	$|T_{\text{0}}-T_{\text{G}}|$
Sunday	$68$	$77$	$-9$	$9$
Monday	$72$	$75$	$-3$	$3$
Tuesday	$74$	$74$	$0$	$0$
Wednesday	$80$	$75$	$5$	$5$
Thursday	$77$	$68$	$9$	$9$
Friday	$71$	$70$	$1$	$1$
Saturday	$70$	$71$	$-1$	$1$

The value $T_0-T_{\text{G}}$ gives more information.

Explanation of Solution

Given:

The bar graph that shows the daily high temperatures for Omak and Genesco during a certain week in June is as follows:

Figure (1)

The temperature in Omak is denoted by $T_{\text{0}}$ and the temperature in Genesco is denoted by $T_{\text{G}}$.

Calculation:

The data from Figure (1) can be tabulated as follows:

	$T_0$	$T_{\text{G}}$
Sunday	$68$	$77$
Monday	$72$	$75$
Tuesday	$74$	$74$
Wednesday	$80$	$75$
Thursday	$77$	$68$
Friday	$71$	$70$
Saturday	$70$	$71$

The absolute value of a real number $x$ is,

$\left|x\right|=\{\begin{array}{c}x\\ -x\end{array}\begin{array}{c}\text{if}x\ge 0\\ \text{if}x<0\end{array}$

So the values of the quantities $T_0-T_{\text{G}}$ and $|T_{\text{0}}-T_{\text{G}}|$ is as follows:

	$T_0$	$T_{\text{G}}$	$T_0-T_{\text{G}}$	$|T_{\text{0}}-T_{\text{G}}|$
Sunday	$68$	$77$	$-9$	$9$
Monday	$72$	$75$	$-3$	$3$
Tuesday	$74$	$74$	$0$	$0$
Wednesday	$80$	$75$	$5$	$5$
Thursday	$77$	$68$	$9$	$9$
Friday	$71$	$70$	$1$	$1$
Saturday	$70$	$71$	$-1$	$1$

The value of $T_0-T_{\text{G}}$ tells about the difference between the quantities $T_{\text{0}}$ and $T_{\text{G}}$. It also tells which of the quantities $T_{\text{0}}$ or $T_{\text{G}}$ is greater.

If $T_0-T_{\text{G}}$ is negative, then $T_{\text{G}}$ is greater than $T_{\text{0}}$.

If $T_0-T_{\text{G}}$ is positive, then $T_{\text{G}}$ is less than $T_{\text{0}}$.

The value of $|T_{\text{0}}-T_{\text{G}}|$ tells us only about the difference between the two quantities $T_{\text{0}}$ and $T_{\text{G}}$.

Therefore, the quantity $T_0-T_{\text{G}}$ gives more information than $|T_{\text{0}}-T_{\text{G}}|$.