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Chapter 1: Problem 78

Suppose you decide to define your own temperature scale with units of \(\mathrm{O}\), using the freezing point \(\left(13^{\circ} \mathrm{C}\right)\) and boiling point \(\left(360^{\circ} \mathrm{C}\right)\) of oleic acid, the main component of olive oil. If you set the freezing point of oleic acid as \(0^{\circ} \mathrm{O}\) and the boiling point as \(100^{\circ} \mathrm{O},\) what is the freezing point of water on this new scale?

Short Answer

Expert verified
The freezing point of water in the new temperature scale O can be found using the conversion formula \(O = m \cdot x_C + b\) based on the given freezing and boiling points of oleic acid in Celsius and the new scale. After calculating the slope (m) and y-intercept (b), we can substitute the freezing point of water in Celsius, \(0^{\circ} \mathrm{C}\), as \(x_C\) in the formula to find its equivalent value in the O scale.

Step by step solution

01

Identify the known conversion points

We are given the information that: Freezing point of oleic acid: \(13^{\circ} \mathrm{C} = 0^{\circ} \mathrm{O}\) Boiling point of oleic acid: \(360^{\circ} \mathrm{C} = 100^{\circ} \mathrm{O}\)
02

Find the linear relationship between Celsius and the new scale

We have two points on the coordinate plane that correspond to the known equivalences between the Celsius and O scales. Let's denote the celsius values as \(x_C\) and the O values as \(O\). Using the slope-intercept equation \(O = m \cdot x_C + b\), we have: Point 1: \( (x_{C1}, O_1) = (13, 0) \); Point 2: \( (x_{C2}, O_2) = (360, 100) \). Now, we'll find the slope and y-intercept for this linear relationship. Slope \(m = \frac{O_2 - O_1}{x_{C2} - x_{C1}} = \frac{100-0}{360-13}\) Use Point 1 to find the y-intercept (b): \( O_1 = m \cdot x_{C1} + b\), solve for \(b\).
03

Establish the conversion formula

Once we have found the values of \(m\) and \(b\), the linear conversion formula between Celsius and the new temperature scale O will be: \(O = m \cdot x_C + b\)
04

Find the freezing point of water in the new scale

We know that the freezing point of water in Celsius is \(0^{\circ} \mathrm{C}\). We will use this value as \(x_C\) in our conversion formula to find the equivalent value, \(O\), in the new temperature scale.

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