Chapter 1.1, Problem 54E

(a)

To determine

To find:The temperature such that the Fahrenheit and Celsius thermometer give the same reading.

Answer to Problem 54E

The Fahrenheit and Celsius thermometer give the same readingat the temperature $-40°\text{C}=-40°\text{F}$ .

Explanation of Solution

Given information:The relationship between the Fahrenheit and Celsius temperature is $F=\frac{9}{5}C+32$ .

Calculation:

Assume that the same reading on the thermometers be $m$ . So, $F=C=m$ .

Substitute $m$ for $F$ and $C$ in the given model.

  $\begin{array}{c}m=\frac{9}{5}m+32\\ 5m=9m+160\\ -4m=160\\ m=-40\end{array}$

Therefore, the Fahrenheit and Celsius thermometer give the same reading at the temperature $-40°\text{C}=-40°\text{F}$ .

(b)

To determine

To draw: The graph of functions $y_1=\frac{9}{5}x+32$ , $y_2=\frac{5}{9}\left(x-32\right)$ and $y_3=x$ in the same viewing window.

Explanation of Solution

Given information:The given functions are $y_1=\frac{9}{5}x+32$ , $y_2=\frac{5}{9}\left(x-32\right)$ and $y_3=x$ .

Graph:

To plot the graph of given function in the same viewing window, follow the steps given below and use graphing calculator.

Step 1: Press “ON” button on the graphing calculator and then press $\boxed{\text{Y}=}$ key. Enter the keystrokes $\boxed{(}\boxed{9}\boxed{÷}\boxed{5}\boxed{)}\boxed{\text{X}}\boxed{+}\boxed{32}$ after the symbol ${\text{Y}}_1$ for the equation $y_1=\frac{9}{5}x+32$ .

Step 2: Press the key $\boxed{\text{ENTER}}$ . Enter the keystrokes $\boxed{(}\boxed{5}\boxed{÷}\boxed{9}\boxed{)}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{32}\boxed{)}$ after the symbol ${\text{Y}}_2$ for the equation $y_2=\frac{5}{9}\left(x-32\right)$ .

Step 3: Press the key $\boxed{\text{ENTER}}$ . Enter the keystrokes $\boxed{\text{X}}$ after the symbol ${\text{Y}}_2$ for the equation $y_3=x$ . Set the window at $\left[-90,90\right]$ by $\left[-60,60\right]$ . Now, press $\boxed{\text{GRAPH}}$ key to produce the graph of the function in the same viewing window as shown below.

  

Figure (1)

Interpretation: From the graph it can be observed that all three lines pass through the point $\left(-40,40\right)$ . At this point the Fahrenheit and Celsius thermometer shows the same temperature.