On a temperature scale 'X', the boiling point of water is 65° X and the freezing point is -15° X. As

On a temperature scale 'X', the boiling point of water is 65° X and the freezing point is -15° X.
Assume that the X scale is linear. The equivalent temperature corresponding to -950 X on the Farenheit scale would be
[JEE 2023]
(a) 630 F
(b) -1120 F
(c) -480 F
(d) -148° F


Concept of Temperature Scale Conversion

Temperature scales can vary, and they may not always align directly with standard scales like Celsius or Fahrenheit. When converting between such scales, the key is understanding the linear relationship between them.

Linear Temperature Scales

A linear temperature scale implies that there is a direct proportional relationship between the temperatures on this scale and those on a standard scale (like Celsius or Fahrenheit). For a given scale, the conversion can generally be described by a linear equation:

\[ T_{\text{standard}} = a \cdot T_{\text{new}} + b \]

where:
- \( T_{\text{standard}} \) is the temperature in the standard scale (e.g., Celsius or Fahrenheit).
- \( T_{\text{new}} \) is the temperature in the new scale.
- \( a \) and \( b \) are constants that define the linear relationship.

Steps for Conversion

1. *Determine Calibration Points:*
- Identify known reference points where the temperature on the new scale corresponds to known temperatures on the standard scale. For example, the freezing and boiling points of water (0°C and 100°C) are commonly used reference points.

2. *Formulate the Linear Relationship:*
- Use the known reference points to set up equations. Typically, you will have two equations based on these reference points, which can be used to solve for the constants \( a \) and \( b \) in the linear equation.

3. *Apply the Conversion Formula:*
- Once you have the linear equation, you can use it to convert temperatures from the new scale to the standard scale. Simply substitute the temperature reading from the new scale into the equation to find the corresponding temperature in the standard scale.

4. *Convert to Desired Scale (if needed):*
- If you need to convert to a different standard scale (e.g., from Celsius to Fahrenheit), use the standard conversion formulas:
- *Celsius to Fahrenheit:*
\[
T_{\text{Fahrenheit}} = \frac{9}{5} \cdot T_{\text{Celsius}} + 32
\]
- *Fahrenheit to Celsius:*
\[
T_{\text{Celsius}} = \frac{5}{9} \cdot (T_{\text{Fahrenheit}} - 32)
\]

Example of Conversion Process

Suppose you have a temperature scale where the freezing point of water is marked as 20 degrees and the boiling point as 120 degrees on this scale. To convert a reading from this scale to Celsius:

1. *Identify Calibration Points:*
- Freezing point: 20 on the new scale = 0°C
- Boiling point: 120 on the new scale = 100°C

2. *Set Up the Linear Relationship:*
- Determine the conversion formula using the given points.

3. *Convert the Temperature:*
- Apply the formula to find the equivalent temperature in Celsius for any given reading on the new scale.

4. *Convert to Another Scale:*
- If needed, convert from Celsius to Fahrenheit using the standard formula.

Conclusion

By understanding and applying these principles, you can effectively convert temperatures between different scales, even if the scales are not initially familiar. Linear relationships simplify the process, making it straightforward to derive accurate conversions.
To solve this problem, we need to convert a temperature from a given scale (X) to the Fahrenheit scale. The given temperature scale X is linear, and we know:

- *Boiling point of water:* 65°X
- *Freezing point of water:* -15°X

These correspond to:

- *Boiling point of water in Celsius:* 100°C
- *Freezing point of water in Celsius:* 0°C

We need to convert -950°X to Fahrenheit.

Step-by-Step Solution

1. *Determine the Linear Relationship Between Scale X and Celsius:*

The relationship between scale X and Celsius is linear, so we can use a formula of the form:
\[
T_{\text{Celsius}} = a \cdot T_{\text{X}} + b
\]

Using the known points:
- At 65°X (boiling point), the temperature is 100°C.
- At -15°X (freezing point), the temperature is 0°C.


It appears there might be a need for a reassessment of the options provided or the accuracy of the provided scale's parameters.
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