Question

Ethylene glycol is the main component in automobile antifreeze. To monitor the temperature of an auto cooling system, you intend to use a meter that reads from 0 to 100 . You devise a new temperature scale based on the approximate melting and boiling points of a typical antifreeze solution \(\left(-45^{\circ} \mathrm{C}\right.\) and \(\left.115^{\circ} \mathrm{C}\right)\). You wish these points to correspond to \(0^{\circ} \mathrm{A}\) and \(100^{\circ} \mathrm{A}\), respectively. a. Derive an expression for converting between \({ }^{\circ} \mathrm{A}\) and \({ }^{\circ} \mathrm{C}\). b. Derive an expression for converting between \({ }^{\circ} \mathrm{F}\) and \({ }^{\circ} \mathrm{A}\). c. At what temperature would your thermometer and a Celsius thermometer give the same numerical reading? d. Your thermometer reads \(86^{\circ} \mathrm{A} .\) What is the temperature in \({ }^{\circ} \mathrm{C}\) and in \({ }^{\circ} \mathrm{F}\) ? e. What is a temperature of \(45^{\circ} \mathrm{C}\) in \({ }^{\circ} \mathrm{A}\) ?

Short answer

The conversion equations are as follows: a. \({}^{\circ} \mathrm{A} = \frac{4}{3} ({}^{\circ}\mathrm{C}) + 60\) b. \({}^{\circ}\mathrm{A} = \frac{40}{27} ({}^{\circ}\mathrm{F}) - \frac{1180}{27}\) c. Thermometers give the same numerical reading at \(-36^{\circ}\mathrm{C}\). d. For a reading of \(86^{\circ}\mathrm{A}\), the temperature is \(19.5^{\circ}\mathrm{C}\) and \(67.1^{\circ}\mathrm{F\). e. A temperature of \(45^{\circ}\mathrm{C}\) corresponds to \(140^{\circ}\mathrm{A}\).

Step-by-step solution

Set up the equation

We will use a linear function to represent the relationship between A-scale (\({}^{\circ}\mathrm{A}\)) and Celsius (\({}^{\circ}\mathrm{C}\)) temperatures. We can set up the equation in the form: \[ {}^{\circ} \mathrm{A} = k({}^{\circ}\mathrm{C}) + b \]

Use known temperature points to find k and b

We know that \(-45^{\circ} \mathrm{C}\) corresponds to \(0^{\circ}\mathrm{A}\), and \(115^{\circ}\mathrm{C}\) corresponds to \(100^{\circ}\mathrm{A}\). Use these points to create two linear equations: \[ 0 = k(-45) + b \] \[ 100 = k(115) + b \]

Solve the system of equations

Solving the system of equations above, we get: \[ k = 4/3 \] \[ b = 60 \]

Write the conversion equation

Using the values of k and b found, the conversion equation from Celsius to A-scale is: \[ {}^{\circ} \mathrm{A} = \frac{4}{3} ({}^{\circ}\mathrm{C}) + 60 \] Now, we will derive an expression for converting \({ }^{\circ} \mathrm{F}\) and \({ }^{\circ} \mathrm{A}\). #b. Derive an expression for converting between \({ }^{\circ} \mathrm{F}\) and \({ }^{\circ} \mathrm{A}\).#

Write the equation to convert Fahrenheit to Celsius

We know the equation to convert between Fahrenheit and Celsius: \[ {}^{\circ}\mathrm{C}=\frac{5}{9} ({}^{\circ}\mathrm{F}-32) \]

Substitute Celsius conversion into A-scale conversion

Replace the Celsius term (\({ }^{\circ}\mathrm{C}\)) in the A-scale conversion equation (derived in part a) with the Fahrenheit conversion: \[ {}^{\circ}\mathrm{A} = \frac{4}{3}\left(\frac{5}{9} ({}^{\circ}\mathrm{F}-32)\right) + 60 \]

Simplify the equation

Simplify the equation to get the expression for converting Fahrenheit to A-scale: \[ {}^{\circ}\mathrm{A} = \frac{40}{27} ({}^{\circ}\mathrm{F}) - \frac{1180}{27} \] #c. At what temperature would your thermometer and a Celsius thermometer give the same numerical reading?#

Set A-scale and Celsius scales equal

Set the A-scale and Celsius conversion equations equal to each other to find the temperature where the numerical reading is the same: \[ {}^{\circ}\mathrm{C} = \frac{4}{3} ({}^{\circ}\mathrm{C}) + 60 \]

Solve for \({ }^{\circ}\mathrm{C}\)

Solve the equation above for \({ }^{\circ}\mathrm{C}\): \[ {}^{\circ}\mathrm{C} = -36 \] So, the two thermometers would give the same numerical reading at \(-36^{\circ}\mathrm{C}\). #d. Your thermometer reads \(86^{\circ} \mathrm{A} .\) What is the temperature in \({ }^{\circ} \mathrm{C}\) and in \({ }^{\circ} \mathrm{F}\)?#

Convert A-scale to Celsius

Use the A-scale to Celsius conversion equation to find the Celsius temperature: \[ {}^{\circ} \mathrm{C} = \frac{3}{4} (86{}^{\circ}\mathrm{A} - 60) \] \[ {}^{\circ} \mathrm{C} = 19.5 \]

Convert Celsius to Fahrenheit

Use the Celsius to Fahrenheit conversion equation to find the Fahrenheit temperature: \[ {}^{\circ}\mathrm{F} = \frac{9}{5} (19.5{}^{\circ}\mathrm{C}) + 32 \] \[ {}^{\circ}\mathrm{F} = 67.1 \] So, when your thermometer reads \(86^{\circ}\mathrm{A}\), the temperature in Celsius is \(19.5^{\circ}\mathrm{C}\) and in Fahrenheit is \(67.1^{\circ}\mathrm{F}\). #e. What is a temperature of \(45^{\circ} \mathrm{C}\) in \({ }^{\circ} \mathrm{A}\)?#

Convert Celsius to A-scale

Use the Celsius to A-scale conversion equation to find the A-scale temperature: \[ {}^{\circ}\mathrm{A} = \frac{4}{3} (45{}^{\circ}\mathrm{C}) + 60 \] \[ {}^{\circ}\mathrm{A} = 140 \] So, a temperature of \(45^{\circ}\mathrm{C}\) corresponds to \(140^{\circ}\mathrm{A}\).

Key concepts

Celsius scale

The Celsius scale is one of the most commonly used temperature scales around the world. It is named after the Swedish astronomer, Anders Celsius, who introduced it in the early 18th century. This scale measures temperature in degrees Celsius (°C) and is based around two major fixed points: the freezing point of water at 0°C and the boiling point of water at 100°C at sea level.

These two points provide a straightforward method for temperature measurement, making it convenient for both everyday and scientific use. The scale is linear, meaning that each degree represents the same temperature change; this makes calculations and conversions involving Celsius straightforward. When converting between the Celsius scale and other temperature scales, such as the Fahrenheit and the new A-scale, it's essential to understand the linear relationship and the consistent increments that these scales represent.

• The freezing point of water is 0°C.
• The boiling point of water is 100°C.
• Each division, or degree, on the Celsius scale is equal to 1/100th of the temperature difference between the freezing and boiling points of water.

This forms the foundation for setting equations to convert from Celsius to other scales, utilizing specific known points and slopes derived from those points. For instance, in customizing a scale like the A-scale, mathematical formulas based on these principles are used to relate Celsius temperatures to the new scale.

Fahrenheit scale

The Fahrenheit scale is another widely recognized temperature measurement system, especially in the United States. It was created by Daniel Gabriel Fahrenheit in the 18th century. This scale is defined by two reference temperature points: the freezing point of water at 32°F and the boiling point at 212°F under standard atmospheric conditions.

Unlike the Celsius scale, the Fahrenheit scale does not align directly with the metric system values for water's freezing and boiling. Consequently, converting between Celsius and Fahrenheit involves a different set of calculations to account for their respective start and end points. The equation to convert Celsius to Fahrenheit is:

• Using the formula: \({}^{ ext{°}}F = \frac{9}{5} \times {}^{ ext{°}}C + 32\)
• This equation suggests that Fahrenheit increments are smaller compared to Celsius, requiring more degrees Fahrenheit to represent the same temperature change.

When relating Fahrenheit to other scales, such as the A-scale developed in the exercise, the fundamental understanding remains similar. One takes the known relation between Celsius and Fahrenheit and integrates it into the conversion formula designed for new scales. Each temperature scale, though distinct in structure, uses these same principles of linearity and fixed reference points.

A-scale temperature

The A-scale temperature is a hypothetical temperature scale created for a specific exercise involving an automotive antifreeze solution. It utilizes reference points based on the characteristics of this solution, where the freezing point of -45°C corresponds to 0°A, and the boiling point of 115°C corresponds to 100°A. This reflects practical uses where custom scales are required for specific materials or processes.

To relate this scale with the Celsius scale, a linear function is established, symbolizing the correlation between these two scales. Using known reference points, we can mathematically express the relationship as:

• Conversion from Celsius to A-scale: \({}^{ ext{°}}A = \frac{4}{3} \, \times {}^{ ext{°}}C + 60\)

This linearity makes it easy to convert temperatures from Celsius to the A-scale using simple multiplication and addition. Typically, these custom scales are used where there's a need for specific calibration to tailor the measurements to unique environmental conditions or material properties, such as monitoring antifreeze in a vehicle.

Overall, creating such scales helps refine the measurement process to better suit specific needs, demonstrating the flexibility and functionality of temperature conversions across different contexts.