Question

A Fahrenheit thermometer registers ${107^ \circ }F$ while a faulty Celsius thermometer registers ${42^ \circ }C$. Find the error in the latter.
A. ${0.37^ \circ }$
B. ${0.33^ \circ }$
C. ${0.87^ \circ }$
D. ${48^ \circ }$

Answer 1

Hint:The temperature in a thermometer can be measured either in Fahrenheit or in Celsius and these two measurements are related to each other as well. The formula relating the temperature of the thermometer in Fahrenheit and Celsius is applied in-order to determine the correct value of the temperature of the Celsius thermometer since it is faulty. This value is compared with the faulty value and the difference between them gives the error in the thermometer measuring the temperature in Celsius.

Formula used:
The equation relating the temperature in Fahrenheit and Celsius is given to be:
${T_c} = \dfrac{5}{9}\left( {{T_f} - 32} \right)$
Where, ${T_c}$ denotes the temperature in Celsius and ${T_f}$ denotes the temperature in Fahrenheit.

Complete step by step answer:
A thermometer is a device that measures the temperature of a body. The thermometer makes use of some measurable quantity which is called as the thermometric property of a substance which changes linearly with temperature. It is a measure or the degree of how hot or cold the body is. Heat is a quantity which flows from higher temperature to lower temperature.

The thermometer can be used to measure the temperature using different scales like the Fahrenheit scale, the Celsius scale and the Kelvin scale. All these thermometers measure the same temperature value but on different scales. Hence we can say that there is a relation between the scales of measurements. Here, the question mentions the Fahrenheit and the Celsius scales only so the first step is to determine the equation relating the two scales. The equation is as follows:
$\dfrac{C}{{100}} = \dfrac{{F - 32}}{{180}}$
This equation is derived from the concept of equating the temperature differences between the highest points of the Celsius and Fahrenheit scales and the lowest points of the scales. This equation is further simplifies in-order to get:
$\dfrac{C}{5} = \dfrac{{F - 32}}{9}$
By cross multiplying the terms and rearranging the terms to make $C$ as the subject we get the common equation relating the two scales.
$C = \dfrac{5}{9}\left( {F - 32} \right)$

We replace $C$ and $F$ with the temperature measured by the Celsius scale as ${T_c}$ and the temperature measured by the Fahrenheit scale as ${T_f}$ respectively we get the equation to be:
${T_c} = \dfrac{5}{9}\left( {{T_f} - 32} \right)$ ------($1$)
We are given that the value of the temperature in the Fahrenheit scale is ${107^ \circ }F$. This value is the correct value of temperature as this is the thermometer that measures temperatures properly. Hence by substituting this value in equation ($1$) we will get the correct value temperature in the Celsius scale which the thermometer is supposed to show.
After substituting the given value we get:
${T_c} = \dfrac{5}{9}\left( {107 - 32} \right)$
We solve the equation to get the value of ${T_c}$:
$ \Rightarrow {T_c} = \dfrac{5}{9} \times 75$
$ \Rightarrow {T_c} = 41.6666$
We round this value to approximately four significant figures or to two decimal places.
$ \Rightarrow {T_c} = 41.66{}^ \circ C$

This is the correct value of temperature in Celsius which means that this is the corresponding value of temperature in the Celsius scale. The thermometer which measures the temperature in Celsius is however said to be faulty and the wrong value is given to be:
${T_{{c_1}}} = 42{}^ \circ C$
The error is given by the difference in these two values of temperatures to find out how much variation has taken place from the correct value of temperature. Hence, the error is given to be:
$\Delta T = {T_{{c_1}}} - {T_c}$
$ \Rightarrow \text{error} = {T_{{c_1}}} - {T_c}$
$ \Rightarrow \text{error} = 42 - 41.66$
$ \therefore \text{error} = {0.33^ \circ }C$
Hence, the error in the Celsius scale is said to be given by ${0.33^ \circ }C$.

Therefore, the correct option is option B.

Additional information: Temperature is a quantity that is said to be defined as the thermal state of the body which decides the direction of flow of heat energy from one body to another when they are placed in thermal contact with each other. A thermometer is named after a Latin word ‘thermo’ which means heat and ‘meter’ which means measurement and the device was invented back in 1612.

Note: The Celsius scale is used everywhere across the globe. This scale is also known as the centigrade scale which was devised based on the degree of freezing and boiling point of water which is \[{0^ \circ }\] and ${100^ \circ }$ respectively. These two scales are derived from the Kelvin scale.