Question

At what temperature is the Fahrenheit scale reading equal to

(a) twice that of the Celsius scale and

(b) half that of the Celsius scale?

Short answer

• a)The Fahrenheit scale reading is equal to twice that of the Celsius scale at$160°Cand320°F$
• b) The Fahrenheit scale reading is equal to half that of the Celsius scale at$-24.61°Cand-12.3°F$

Step-by-step solution

Write the given data

• i)The Fahrenheit scale equal to twice that of the Celsius scale:$T_F=2T_C$
• ii) The Fahrenheit scale equal to half that of the Celsius scale:$T_F=\frac{1}{2}{T}_C$

Understanding the concept of temperature conversion

By using the relation between the Fahrenheit and Celsius scales, we will find all the answers.

Formula:

Using the relation between the Fahrenheit and Celsius scales as follows:
$T_F=\frac{9}{5}{T}_C+32.......\left(i\right)$

a) Calculation of the temperature in the scale reading

In the first case, it is given that the Fahrenheit scale reading is equal to twice that of the Celsius scale that is given as:

$T_F=2T_C$ ….. (ii)

Using condition (ii) and equation (i), and solve for the Celsius temperature as:
$$\begin{gathered} 2T_C=\frac{9}{5}{T}_C+321 \\ 2T_C-\frac{9}{5}{T}_C=32 \\ \left(2-\frac{9}{5}\right)T_C=32 \\ {T}_C=160° \end{gathered}$$

Hence, the Fahrenheit scale using equation (ii) is given as:
$$\begin{gathered} T_F=2\times 160 \\ =320°F \end{gathered}$$

At $T_C=160°C$and $T_F=320°F$the Fahrenheit scale reading is equal to twice that of the Celsius scale.

b) Calculate the temperature in the scale reading

In the second case, it is given that the Fahrenheit scale reading is equal to half of the Celsius scale that is

$T_F=\frac{1}{2}{T}_C$ …. (iii)

Using condition (iii) and equation (i), we get the Celsius temperature as:

$$\begin{gathered} \frac{1}{2}{T}_C=\frac{9}{5}{T}_C+32 \\ \frac{1}{2}{T}_C-\frac{9}{5}{T}_C=32 \\ \left(\frac{1}{2}-\frac{9}{5}\right)T_C=32 \\ {T}_C=-24.61°F \end{gathered}$$

Hence, the Fahrenheit scale using condition (iii) is given as:
$$\begin{gathered} T_F=\frac{1}{2}\times (-24.61) \\ =-12.3°F \end{gathered}$$

At $T_C=-24.61°C$and $T_F=-12.3°F$the Fahrenheit scale reading is equal to half of the Celsius scale.