Question

A Centigrade and a Fahrenheit thermometer are dipped in boiling water. The water temperature is lowered until the Fahrenheit thermometer registers ${{140}^{{}^\circ }}$. What is the fall temperature as registered by the Centigrade thermometer?
A. ${{30}^{{}^\circ }}$
B. ${{40}^{{}^\circ }}$
C. ${{60}^{{}^\circ }}$
D. ${{80}^{{}^\circ }}$

Answer 1

Hint: Use the relation between Fahrenheit temperature and Centigrade temperature and obtain a relation between their corresponding changes. Then use the values of temperature of boiling water and the temperature registered by Fahrenheit thermometer after water temperature is lowered to obtain the temperature as registered by the Centigrade thermometer.
Formula used:
${{T}_{F}}=\dfrac{9}{5}{{T}_{C}}+32$

Complete answer:
We first assume that temperature measured in Centigrade is denoted by ${{T}_{C}}$ and temperature measured in Fahrenheit is denoted by ${{T}_{F}}$. The temperature in centigrade is related to temperature in Fahrenheit as
${{T}_{F}}=\dfrac{9}{5}{{T}_{C}}+32$
The change in Fahrenheit temperature can be related to Change in Centigrade temperature by
$\Delta {{T}_{F}}=\dfrac{9}{5}\Delta {{T}_{C}}$ ……. (1)
It is given that Centigrade and a Fahrenheit thermometer are dipped in boiling water. The temperature of boiling water is ${{100}^{{}^\circ }}C$. Therefore, temperature measured by Centigrade thermometer is ${{100}^{{}^\circ }}C$ while temperature measured by the Fahrenheit thermometer is $\dfrac{9}{5}\times 100+32={{212}^{{}^\circ }}F$. We denote these by ${{T}_{F}}_{1}$ and ${{T}_{F}}_{1}$ respectively.
Then, water temperature is lowered to ${{140}^{{}^\circ }}F$ as measured by Fahrenheit thermometer. The change in temperature as measured by Fahrenheit thermometer is
$\Delta {{T}_{F}}={{T}_{F}}_{2}-{{T}_{F1}}={{140}^{{}^\circ }}F-{{212}^{{}^\circ }}F$
That is, $\Delta {{T}_{F}}=-{{72}^{{}^\circ }}F$
We assume that the temperature denoted by the Centigrade thermometer is ${{T}_{C}}_{2}$. Then change in temperature as registered by the Centigrade thermometer is $\Delta T_C=T_{C2}-T_{C1}$
Substituting these values in equation (1), we get
$-{{72}^{{}^\circ }}F=\dfrac{9}{5}({{T}_{C}}_{2}-{{T}_{C}}_{1})$
Solving this, we have
$\Rightarrow {{T}_{C}}_{2}-{{T}_{C}}_{1}=-{{40}^{{}^\circ }}C$
This is the change in temperature as registered by the Centigrade thermometer.

So, the correct answer is “Option B”.

Note:
Temperature of boiling water is ${{100}^{{}^\circ }}C$. Students must learn this value as this may not be given in the question.
The value of Fahrenheit temperature can also be converted, as measured after the water temperature is lowered, directly into Centigrade temperature and then the difference can be calculated.