Question

On a new scale of temperature (which is linear) and called the W scale, the freezing and boiling points of water are \[39^\circ {\text{W}}\] and \[239^\circ {\text{W}}\] respectively. What will be the temperature on the new scale, corresponding to the temperature of \[39^\circ {\text{C}}\] on the Celsius scale?
A. \[78^\circ {\text{W}}\]
B. \[117^\circ {\text{W}}\]
C. \[200^\circ {\text{W}}\]
D. \[139^\circ {\text{W}}\]

Answer 1

Hint:Use the formula for determining the temperature on an unknown scale corresponding to the temperature on the unknown scale. The formula includes the lower fixed point and upper fixed point on the known and unknown scales for the same value of temperature. Rewrite this formula for the known Celsius scale and unknown W scale and calculate the required value of temperature.

Formula used:
The formula for determining the temperature on an unknown scale is given by
\[\dfrac{{\left( {{\text{UFP on unknown scale}}} \right) - \left( {{\text{LFP on unknown scale}}} \right)}}{{\left( {{\text{UFP on known scale}}} \right) - \left( {{\text{LFP on known scale}}} \right){\text{ }}}} = \dfrac{{t - t\,{\text{on known scale}}}}{{t\,{\text{on known scale}}}}\]
                                                                 …… (1)
Here, UFP is the upper fixed point, LFP is the lower fixed point and $t$ is the temperature in the unknown scale corresponding to the LFP of the known scale.

Complete step by step answer:
The new scale is called W scale. We have given that on the new scale which is linear, the freezing and boiling points of the water are \[39^\circ {\text{W}}\] and \[239^\circ {\text{W}}\] respectively.
\[{\text{LFP on W scale}} = 39^\circ {\text{W}}\]
\[\Rightarrow{\text{UFP on W scale}} = 239^\circ {\text{W}}\]
We know that the freezing point and boiling point of water on the Celsius scale are and \[100^\circ {\text{C}}\] respectively.
\[{\text{LFP on Celsius scale}} = 0^\circ {\text{C}}\]
\[\Rightarrow{\text{UFP on Celsius scale}} = 100^\circ {\text{C}}\]

We have asked to calculate the temperature corresponding to \[39^\circ {\text{C}}\] on the W scale. We can calculate the temperature corresponding to \[39^\circ {\text{C}}\] on W scale using equation (1). Rewrite equation (1) for known Celsius scale and unknown W scale.
\[\dfrac{{\left( {{\text{UFP on W scale}}} \right) - \left( {{\text{LFP on W scale}}} \right)}}{{\left( {{\text{UFP}} - {\text{LFP}}} \right){\text{ on Celsius scale}}}} = \dfrac{{t - t\,{\text{on Celsius scale}}}}{{t{\text{ on Celsius scale}}}}\]

Substitute \[239^\circ {\text{W}}\] for UFP on W scale, \[39^\circ {\text{W}}\] for LFP on W scale, \[100^\circ {\text{C}}\] for UFP on Celsius scale, \[0^\circ {\text{C}}\] for LFP on Celsius scale and \[39^\circ {\text{C}}\] for t on Celsius scale in the above equation.
\[\dfrac{{\left( {239^\circ {\text{W}}} \right) - \left( {39^\circ {\text{W}}} \right)}}{{\left( {100^\circ {\text{C}}} \right) - \left( {0^\circ {\text{C}}} \right)}} = \dfrac{{t - 39^\circ {\text{C}}}}{{39^\circ {\text{C}}}}\]

\[ \Rightarrow \dfrac{{\left( {200^\circ {\text{W}}} \right) \times \left( {39^\circ {\text{C}}} \right)}}{{100^\circ {\text{C}}}} = t - 39^\circ {\text{C}}\]
\[ \Rightarrow 78 = t - 39\]
\[ \Rightarrow t = 78 + 39\]
\[ \therefore t = 117^\circ {\text{W}}\]
Therefore, the required temperature on the W scale is \[117^\circ {\text{W}}\].

Hence, the correct option is B.

Note: One can solve the same question by another method. One can use the division method to determine the required temperature in the unknown scale. Determine the number of divisions on the known and unknown scale corresponding to one degree temperature on both the scales and calculate the required temperature in the unknown scale.