Suppose you were dissatisfied with both Celsius and Fahrenheit units and wanted to design your own temperature scale based on ethyl alcohol (ethanol). On the Celsius scale, ethanol has a melting point of -117.3 °C and a boiling point of 78.5 °C, but on your new scale calibrated in units of degrees ethanol, °E, you define ethanol to melt at 0 °E and boil at 200 °E. (d) What is normal human body temperature (98.6 °F) on the ethanol scale?

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Convert the normal human body temperature from Fahrenheit to Celsius using the formula: \( T_{\text{C}} = \frac{5}{9}(T_{\text{F}} - 32) \).

Identify the relationship between the Celsius scale and the ethanol scale. Ethanol melts at -117.3 °C (0 °E) and boils at 78.5 °C (200 °E).

Calculate the conversion factor between the Celsius and ethanol scales by finding the difference in temperature for both scales: \( 78.5 - (-117.3) \) °C corresponds to \( 200 - 0 \) °E.

Use the conversion factor to create a linear equation relating °C to °E: \( T_{\text{E}} = m(T_{\text{C}} - (-117.3)) \), where \( m \) is the conversion factor.

Substitute the Celsius temperature of the human body into the linear equation to find the temperature in degrees ethanol, °E.

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주요 개념

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Temperature Scales

Temperature scales, such as Celsius and Fahrenheit, provide a way to quantify thermal energy. Each scale has defined reference points; for instance, the freezing and boiling points of water are 0 °C and 100 °C in Celsius, and 32 °F and 212 °F in Fahrenheit. Understanding how to convert between these scales is essential for solving temperature-related problems.

Linear Interpolation

Linear interpolation is a mathematical method used to estimate unknown values that fall within a range of known values. In this context, it can be applied to convert temperatures from one scale to another by establishing a linear relationship between the defined points of the scales. For example, knowing the melting and boiling points of ethanol allows us to create a linear equation to find other temperatures.

Conversion Formulas

Conversion formulas are equations that allow for the transformation of values from one unit to another. In this case, to convert Fahrenheit to the new ethanol scale, one must first convert Fahrenheit to Celsius and then apply the linear relationship established for the ethanol scale. Understanding how to derive and apply these formulas is crucial for accurate temperature conversions.