Question

The formula for converting from Celsius to Fahrenheit temperatures is $F = \dfrac{9}{5}C + 32$ what is the inverse of this formula? Is the inverse a function? What is the Celsius temperature that corresponds to $27^\circ F$ ?

Answer 1

In order to solve this question we need to understand the temperature and its measurement. Temperature is defined as a parameter to measure how hot or cold a body is. And heat is defined as energy which flows only due to temperature difference. There are three basic scales to measure the temperature of a body, first is Degree Fahrenheit, second is Degree Celsius, and third is Kelvin. Inverse of function exist when for pre-image in the range of function there is image in domain of the function.

Complete step by step answer:
Let the Fahrenheit scale be denoted as $F$ and Celsius scale be denoted as $C$.So the conversion from Celsius to Fahrenheit is,
$F = \dfrac{9}{5}C + 32 \to (i)$
For inverse of this formula, we would find conversion from Fahrenheit to Celsius as,
$F - 32 = \dfrac{9}{5}C$
$\Rightarrow (F - 32)\dfrac{5}{9} = C$
$\Rightarrow C = \dfrac{5}{9}(F - 32) \to (ii)$
This is the inverse of function (i) because for every F there is value in C so from the definition of inverse this function (ii) is inverse of (i).

Numerical Part: Temperature given in Fahrenheit is, $T = 27^\circ F$.Let the temperature in Celsius be ${T_1}$.So putting in formula (ii) we get,
${T_1} = \dfrac{5}{9}(T - 32)$
$\Rightarrow {T_1} = \dfrac{5}{9}(27 - 32)$
$\Rightarrow {T_1} = \dfrac{5}{9}( - 5)$
$\therefore {T_1} = - 2.78^\circ C$

So Celsius temperature is, $- 2.78^\circ C$.

Note: It should be remembered that out of three scales, kelvin, Celsius and Fahrenheit, Kelvin is SI unit for temperature measure it is due to fact that Kelvin measurement is independent of working substance in thermometer, it is always positive however Celsius and Fahrenheit both could be negative and thus create error in measurement of temperature, not only at macro scales but at micro also. $0K$ is considered as absolute zero.