Question

At what temperature, do the Celsius and Fahrenheit scales show the same reading but with opposite signs?
$\begin{align}
  & \text{A}\text{. }11.44 \\
 & \text{B}\text{. }20 \\
 & \text{C}\text{. }40 \\
 & \text{D}\text{. }10 \\
\end{align}$

Answer 1

Hint: There are several well-defined scales of measurement of temperature in different units. These scale values can be interconverted into one another. For finding the temperature where Celsius and Fahrenheit scales show the same reading but with opposite signs, we will use the conversion equation from Fahrenheit to Celsius scale.

Complete step by step answer:
The degree Celsius is a unit of measuring temperature on the Celsius scale. Celsius scale is a temperature scale, called a centigrade scale. The degree Celsius, represented as$^{\text{o}}\text{C}$, refers to a particular temperature on the Celsius scale or a unit to specify the difference between two temperatures.
Fahrenheit is a temperature scale that specifies the boiling point of water at ${{212}^{\text{o}}}\text{F}$and the freezing point of water at${{32}^{\text{o}}}\text{F}$. It uses the degree Fahrenheit, represented as$^{\text{o}}\text{F}$, as its measuring unit to indicate a specific temperature as well as to denote the difference between two temperatures.
Zero degrees Celsius is defined as $273.15K$. One degree Celsius, centigrade temperature, is equal to one Kelvin; therefore, boiling point of water is equal to$273.15+100=373.15\text{ Kelvin}$. The Fahrenheit temperature range is based on setting the freezing point of water at$32\text{ degrees}$, and boiling point to$212\text{ degrees}$. This means that boiling and freezing points are $180\text{ degrees}$apart. Absolute zero is defined as $-459.67{{\text{ }}^{\text{o}}}\text{F}$
Celsius to Fahrenheit scale:
\[^{\text{o}}\text{F=}\left( \dfrac{\text{9}}{\text{5}} \right){{\times }^{\text{o}}}\text{C+32}\]
We have to calculate that value of temperature, where Celsius and Fahrenheit scales show the same reading but with opposite sign,
It means,
$^{\text{o}}\text{F = }{{\text{-}}^{\text{o}}}\text{C}$
We will use the algebraic equation to determine the value
Put $^{\text{o}}\text{F = }-{{\text{ }}^{\text{o}}}\text{C}$in conversion equation,
\[\begin{align}
  & -{{\text{ }}^{\text{o}}}\text{C =}{{\text{ }}^{\text{o}}}\text{C }\!\!\times\!\!\text{ }\left( \dfrac{\text{9}}{\text{5}} \right)\text{+32} \\
 & -{{\text{ }}^{\text{o}}}\text{C}-{{\text{ }}^{\text{o}}}\text{C}\left( \dfrac{\text{9}}{\text{5}} \right)\text{ = 32} \\
 & -\left( \dfrac{\text{14}}{\text{5}} \right){{\text{ }\!\!\times\!\!\text{ }}^{\text{o}}}\text{C = 32} \\
 & ^{\text{o}}\text{C =}-\dfrac{\text{80}}{7} \\
\end{align}\]
\[\left| ^{\text{o}}\text{C} \right|\text{ = }\left| ^{\text{o}}\text{F} \right|\text{ = 11}\text{.44}\]
The value where the Celsius and Fahrenheit scales coincide but with opposite sign is $11.44$
Hence, the correct option is A.

Note: Celsius scale, or centigrade scale, is a temperature scale that is based on the freezing point of water at ${{\text{0}}^{\text{o}}}\text{C}$and the boiling point of water at$\text{10}{{\text{0}}^{\text{o}}}\text{C}$. Fahrenheit scale is a temperature scale that is based on the freezing point of water at $\text{3}{{\text{2}}^{\text{o}}}\text{F}$and the boiling point of water at$\text{21}{{\text{2}}^{\text{o}}}\text{F}$.