Suppose you design a new thermometer called the X thermometer. On the X scale, the boiling point of water is 130 °X, and the freezing point of water is 10 °X. At what temperature are the readings on the Fahrenheit and X thermometers the same?

Verified step by step guidance

insert step 1> Determine the relationship between the X scale and the Celsius scale. The freezing point of water is 0 °C and corresponds to 10 °X, while the boiling point of water is 100 °C and corresponds to 130 °X.

insert step 2> Calculate the conversion factor between the X scale and the Celsius scale. Use the formula: (X - 10) / (130 - 10) = (C - 0) / (100 - 0).

insert step 3> Solve the equation from step 2 to express the temperature in Celsius (C) in terms of the temperature in X (X).

insert step 4> Use the relationship between Celsius and Fahrenheit: F = (9/5)C + 32. Substitute the expression for C from step 3 into this equation to express F in terms of X.

insert step 5> Set the expression for F equal to X to find the temperature at which the Fahrenheit and X readings are the same. Solve for X.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Temperature Scales

Temperature scales, such as Celsius, Fahrenheit, and Kelvin, provide a way to quantify thermal energy. Each scale has specific reference points; for example, water freezes at 0 °C and boils at 100 °C in Celsius, while in Fahrenheit, these points are 32 °F and 212 °F, respectively. Understanding how these scales relate to one another is crucial for converting temperatures and solving problems involving different units.

Linear Interpolation

Linear interpolation is a method used to estimate unknown values that fall within a range of known values. In the context of temperature scales, it allows us to determine the equivalent temperature on one scale based on known reference points on another scale. For the X thermometer, knowing the freezing and boiling points of water enables us to create a linear relationship to find other temperatures.

Equation of Temperature Conversion

The equation for converting between temperature scales is essential for solving problems involving different units. For Fahrenheit (F) and Celsius (C), the conversion is F = (9/5)C + 32. In this case, we need to derive a similar equation for the X thermometer based on its defined freezing and boiling points, allowing us to find the temperature at which both the Fahrenheit and X scales read the same.

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Open Question

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