Problem 78 - Introductory and Intermediate Algebra for College Students 4th

Question

The relationship between Celsius temperature, \(C,\) and Fahrenheit temperature, \(F\), can be described by a linear equation in the form \(F=m C+b .\) The graph of this equation contains the point \((0,32):\) Water freezes at \(0^{\circ} \mathrm{C}\) or at \(32^{\circ} \mathrm{F}\). The line also contains the point \((100,212):\) Water boils at \(100^{\circ} \mathrm{C}\) or at \(212^{\circ} \mathrm{F}\). Write the linear equation expressing Fahrenheit temperature in terms of Celsius temperature.

Step-by-Step Solution

Answer

The linear equation expressing Fahrenheit temperature in terms of Celsius temperature is \(F=1.8C+32\).

Step 1: Identify the Points

The problem gives two points: (0,32) and (100,212) that are on the line. These points correspond to the temperatures at which water freezes and boils, respectively. In each point, the first value corresponds to the Celsius temperature (C), while the second corresponds to Fahrenheit temperature (F).

Step 2: Calculate Slope

The slope is calculated using the formula \(m=(y_2-y_1)/(x_2-x_1)\). Substituting the given points into this equation: \(m=(212−32)/(100−0)=180/100=1.8\).

Step 3: Find The Y-Intercept

We already know that the y-intercept (b) is where x=0. From our points, we see when \(C=0\), \(F=32\). So, \(b = 32\).

Step 4: Write the Final Equation

Now we substitute 'm' and 'b' with the values we obtained. This will give us the linear equation that express Fahrenheit temperature in terms of Celsius temperature. It is \(F=1.8C+32\).

Key Concepts

Celsius Temperature

The Celsius temperature scale is a system of measuring temperature, commonly used around the world. It is part of the metric system and is easy to understand:

• Water freezes at 0°C.
• Water boils at 100°C.

This scale is especially useful because it is based on physical properties of water, which makes it intuitive. When we convert Celsius to Fahrenheit or other temperature scales, we use established formulas to maintain consistency and allow accurate scientific measurements across different regions.

Fahrenheit Temperature

Fahrenheit is a temperature scale that is widely used in the United States. It differs from Celsius in terms of its fixed points:

• Water freezes at 32°F.
• Water boils at 212°F.

These points can sometimes make conversions a bit tricky, but they are based on the practicality within specific regions. The equation relating Fahrenheit and Celsius is essential for converting between these scales. Knowing this helps in various scientific and everyday contexts.

Slope Calculation

When dealing with linear equations, calculating the slope is a crucial step. The slope is a measure of how steep a line is. It can be determined using two points on a line, with the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In our temperature conversion example:

• Point 1: (0, 32)
• Point 2: (100, 212)

Substituting these values gives:
\[ m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \] This value, 1.8, indicates how the Fahrenheit temperatures increase as the Celsius temperatures increase.

Graphing Linear Equations

Understanding the graph of a linear equation is important in visualizing relationships, such as temperature conversions. The equation we derived is \( F = 1.8C + 32 \).
When we graph this equation, it is essential to plot key points:

• The y-intercept, where the line crosses the y-axis: (0, 32).
• The boiling point of water: (100, 212).

These points help us to draw an accurate line. The slope tells us the rise over run, meaning for every Celsius degree increase, Fahrenheit increases by 1.8. Visualizing it this way helps see the steady relationship between these two scales.