4.3 Q-19

Question Statement

Plot the Celsius (C) and Fahrenheit (F) temperature scales on the horizontal axis and the vertical axis, respectively. Draw the line joining the freezing point and the boiling point of water. Then, find an equation that expresses the Fahrenheit temperature $F$ in terms of the Celsius temperature $C$.

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Background and Explanation

In this problem, we are dealing with the relationship between Celsius and Fahrenheit temperature scales. The freezing point of water is $0^\circ C$ (which corresponds to $32^\circ F$), and the boiling point of water is $100^\circ C$ (which corresponds to $212^\circ F$).

To find the equation of the line that connects these points, we will use the point-slope form of the equation of a line: $F - F_1 = m(C - C_1)$
where:

• $(C_1, F_1)$ are the coordinates of one point (e.g., the freezing point),
• $m$ is the slope of the line,
• $C$ and $F$ represent the variables along the horizontal and vertical axes.

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Solution

1. Choose the Points:
   We know the freezing and boiling points of water:

   • Freezing point: $(0^\circ C, 32^\circ F)$,
   • Boiling point: $(100^\circ C, 212^\circ F)$.

2. Calculate the Slope $m$:
   The slope of the line connecting the two points is given by: $m = \frac{F_2 - F_1}{C_2 - C_1}$
   Substituting the values of the boiling and freezing points: $m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = \frac{9}{5}$
   So, the slope of the line is $\frac{9}{5}$.

3. Write the Equation Using Point-Slope Form:
   Using the point-slope form $F - F_1 = m(C - C_1)$, and substituting $F_1 = 32$, $C_1 = 0$, and $m = \frac{9}{5}$, we get: $F - 32 = \frac{9}{5}(C - 0)$
   Simplifying: $F - 32 = \frac{9}{5}C$
   So, the equation becomes: $F = \frac{9}{5}C + 32$
   This is the equation that gives the Fahrenheit temperature $F$ in terms of the Celsius temperature $C$.

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Key Formulas or Methods Used

• Slope Formula:
  $m = \frac{F_2 - F_1}{C_2 - C_1}$

• Point-Slope Form of a Line:
  $F - F_1 = m(C - C_1)$

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Summary of Steps

1. Identify the freezing point $(0^\circ C, 32^\circ F)$ and boiling point $(100^\circ C, 212^\circ F)$.
2. Calculate the slope $m = \frac{9}{5}$ using the formula $m = \frac{F_2 - F_1}{C_2 - C_1}$.
3. Write the equation of the line in point-slope form:
   $F - 32 = \frac{9}{5}C$
4. Simplify to get the equation:
   $F = \frac{9}{5}C + 32$
   This is the equation for Fahrenheit in terms of Celsius.