Question

Question:(II) Determine the temperature at which the Celsius and Fahrenheit scales give the same numerical reading .\(\left( {{T_{\bf{C}}} = {T_{\bf{F}}}} \right)\)

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Short answer

The temperature at which the Celsius and Fahrenheit readings are the same is \( - 40^\circ {\rm{C}} = - 40^\circ {\rm{F}}\).

Step-by-step solution

Understanding the scales of the temperature

The temperature is mainly measured on the scales: Kelvin, Fahrenheit, and Celsius.In this problem, the temperature value provides the same Celsius and Fahrenheit readings. Then, evaluate the required temperature. The unit of temperature can be expressed in degrees Centigrade and Fahrenheit.

Determination of the temperature at which the Celsius and the Fahrenheit have the same readings

It is given that the temperature readings of Celsius and Fahrenheit are the same. It can be expressed as

\({T_{\rm{F}}} = {T_{\rm{C}}}\).

The conversion of the temperature from Celsius to Fahrenheit can be expressed as follows:

\({T_{\rm{F}}} = \frac{9}{5}{T_{\rm{C}}} + 32\)

Substitute the values in the above equation.

\(\begin{aligned}{c}{T_{\rm{C}}} &= \frac{9}{5}{T_{\rm{C}}} + 32\\{T_{\rm{C}}} - \frac{9}{5}{T_{\rm{C}}} &= 32\\ - \frac{4}{5}{T_{\rm{C}}} &= 32\\{T_{\rm{C}}} &= - 40\end{aligned}\)

This can be expressed as follows:

\(\begin{aligned}{c}{T_{\rm{C}}} &= - 40^\circ {\rm{C}}\\{T_{\rm{F}}} &= - 40^\circ {\rm{F}}\end{aligned}\)

Thus, the temperature at which the Celsius and Fahrenheit readings are the same is \( - 40^\circ {\rm{C}} = - 40^\circ {\rm{F}}\).