Question

On the planet Xgnu, the natives have 14 fingers. On the official Xgnuese temperature scale ( \(^{\circ} \mathrm{X}\) ), the boiling point of water (under an atmospheric pressure similar to Earth's) is \(140^{\circ} \mathrm{X},\) whereas it freezes at \(14^{\circ} \mathrm{X}\). Derive the relationship between \(^{\circ} \mathrm{X}\) and \(^{\circ} \mathrm{C}\).

Short answer

The relationship between the Xgnuese temperature scale (°X) and the Celsius temperature scale (°C) can be derived using the equation: \(T_C = \frac{100^{\circ} \mathrm{C}}{126^{\circ} \mathrm{X}} (T_X - 14^{\circ} \mathrm{X})\), where \(T_C\) represents the Celsius temperature and \(T_X\) represents the Xgnuese temperature.

Step-by-step solution

Convert the boiling and freezing points to Celsius

To derive the relationship between the °X and °C systems, we first convert the boiling and freezing points of water on the Xgnu planet to Celsius. The boiling point of water in Celsius is \(100^{\circ} \mathrm{C}\). The freezing point of water in Celsius is \(0^{\circ} \mathrm{C}\).

Determine the difference between the boiling and freezing points in both systems

Next, we need to determine the difference between the boiling and freezing points in both the °X and °C systems. In the Xgnuese temperature scale: Boiling point - Freezing point = \(140^{\circ} \mathrm{X} - 14^{\circ} \mathrm{X} = 126^{\circ} \mathrm{X}\) In the Celsius temperature scale: Boiling point - Freezing point = \(100^{\circ} \mathrm{C} - 0^{\circ} \mathrm{C} = 100^{\circ} \mathrm{C}\)

Calculate the ratio between the two temperature scales

Now that we know the differences between boiling and freezing points in both systems, we can calculate the ratio between the two temperature scales. Ratio = \(\frac{Difference\ in\ °C}{Difference\ in\ °X} = \frac{100^{\circ} \mathrm{C}}{126^{\circ} \mathrm{X}}\)

Write the linear equation relating °X and °C

Now that we have our ratio, we must relate the Xgnuese and Celsius scales in terms of the freezing and boiling points. We can utilize these as reference points to create the linear equation. Let's set \(T_C\) as the Celsius temperature and \(T_X\) as the °X temperature. We start by using the freezing point as a reference point: \(T_C - 0^{\circ} \mathrm{C} = \frac{100^{\circ} \mathrm{C}}{126^{\circ} \mathrm{X}} (T_X - 14^{\circ} \mathrm{X})\) Now, by solving for \(T_C\), we get the equation relating both temperature scales: \(T_C = \frac{100^{\circ} \mathrm{C}}{126^{\circ} \mathrm{X}} (T_X - 14^{\circ} \mathrm{X})\) This is the relationship between the Xgnuese and Celsius temperature scales.

Key concepts

Xgnuese temperature scale

The Xgnuese temperature scale is an imaginary unit of measurement used to gauge temperature on the fictional planet Xgnu. This unique scale gives us insight into how temperature systems can vary across different contexts. In this scale, the boiling point of water is set at \(140^{\circ} \mathrm{X}\) while the freezing point is at \(14^{\circ} \mathrm{X}\).

Understanding temperature scales globally allows us to see the parallels and differences among various measuring systems. This particular scale coincides with the number of fingers the inhabitants of Xgnu have—14. Such considerations remind us that cultural elements often influence scientific conventions.

boiling point of water

The boiling point of water is a critical phenomenon and is often used as a standard point in temperature scales. On Earth, it is universally recognized as \(100^{\circ} \mathrm{C}\) under normal atmospheric pressure. This means it is the temperature at which water transitions from liquid to gas.

On the Xgnuese temperature scale, however, water boils at a significantly different value: \(140^{\circ} \mathrm{X}\).

Analyzing boiling points across different scales helps in understanding the underlying conversions and relationships between different temperature measurement units. It is an essential step in devising accurate scientific models when investigating extraterrestrial conditions.

freezing point of water

The freezing point of water is another key reference used widely across temperature scales. In Earth's Celsius scale, water freezes at \(0^{\circ} \mathrm{C}\). This provides a baseline for both everyday and scientific temperature measurements.

In the Xgnuese temperature scale, this freezing point is marked at \(14^{\circ} \mathrm{X}\).

By comparing such fixed points across different systems, it allows us to convert and compare temperatures accurately. These fixed markers on various scales cement their importance in formulating a clear picture of the difference in how temperature is perceived across civilizations, whether real or fictional.

Celsius temperature scale

The Celsius temperature scale is one of the most widely used temperature measurement systems around the world. Defined by the freezing point at \(0^{\circ} \mathrm{C}\) and the boiling point at \(100^{\circ} \mathrm{C}\), this scale is vital for everyday use as well as scientific research.

This coherent, logical scale offers intuitive reference points that match the physical states of water. By reducing the complexity of more ancient systems, it provides clarity and ease in temperature conversion. The simplicity of Celsius allows for easy translation into other scales, facilitating communication and understanding across different scientific communities.

linear equation

A linear equation involving temperature scales is fundamental in understanding the relationship between different systems. Such equations help us convert measurements from one scale to another with ease and accuracy.

In the context of the Xgnuese and Celsius scales, we derived the equation:\[ T_C = \frac{100^{\circ} \mathrm{C}}{126^{\circ} \mathrm{X}} (T_X - 14^{\circ} \mathrm{X}) \]where \(T_C\) is the temperature in Celsius and \(T_X\) is the temperature in Xgnuese.

This equation leverages the known boiling and freezing points on each scale to establish a proportional relationship. The use of linear equations in this way simplifies the conversion process, making it accessible and understandable for students and scientists. Linear equations enable precision and consistency in temperature analysis and interpretation.