Problem 61
Question
GENERAL: Temperature On the Fahrenheit temperature scale, water freezes at \(32^{\circ}\) and boils at \(212^{\circ} .\) On the Celsius (centigrade) scale, water freezes at \(0^{\circ}\) and boils at \(100^{\circ}\). a. Use the two (Celsius, Fahrenheit) data points \((0,32)\) and \((100,212)\) to find the linear relationship \(y=m x+b\) between \(x=\) Celsius temperature and \(y=\) Fahrenheit temperature. b. Find the Fahrenheit temperature that corresponds to \(20^{\circ}\) Celsius.
Step-by-Step Solution
Verified Answer
The Fahrenheit temperature for 20°C is 68°F.
1Step 1: Understand the Problem
We need to find the linear equation that relates Celsius and Fahrenheit using the given points (0,32) and (100,212), and then use that equation to determine the Fahrenheit equivalent of 20°C.
2Step 2: Determine the Slope, m
The slope, \(m\), of a line is given by the change in \(y\) over the change in \(x\). Use the points \((x_1, y_1) = (0, 32)\) and \((x_2, y_2) = (100, 212)\).\\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \]
3Step 3: Find the Intercept, b
The equation of the line is \(y = mx + b\). Using one of the points, substitute to find \(b\). Use \((0, 32)\):\\[ 32 = 1.8 \times 0 + b \] \\[ b = 32 \]
4Step 4: Write the Linear Equation
Now that we have both \(m\) and \(b\), we can write the linear equation relating Fahrenheit \((y)\) and Celsius \((x)\):\\[ y = 1.8x + 32 \]
5Step 5: Convert 20°C to Fahrenheit
Substitute \(x = 20\)°C into the equation \(y = 1.8x + 32\) to find the corresponding Fahrenheit temperature:\\[ y = 1.8 \times 20 + 32 = 36 + 32 = 68 \]
Key Concepts
Temperature ConversionSlope and InterceptMathematical Modeling
Temperature Conversion
The relationship between Celsius and Fahrenheit scales is crucial to understand how different temperature systems work. Water, a common reference in temperature scales, freezes at 0°C and boils at 100°C. On the Fahrenheit scale, the same points are 32°F and 212°F, respectively. These differences in freezing and boiling points create a linear relationship between the two scales.
We can express this relationship as a linear equation where Celsius is represented by the variable \(x\) and Fahrenheit by \(y\), forming a line equation of the form \(y = mx + b\). Here, \(m\) is the slope indicating how many Fahrenheit degrees a Celsius degree corresponds to, and \(b\) is the y-intercept where the line crosses the y-axis.
The distinct freezing and boiling values provide us two points, namely \((0, 32)\) and \((100, 212)\), which are used to derive the linear equation. Understanding this conversion is immensely helpful in everyday life for weather readings, cooking, and scientific experiments.
We can express this relationship as a linear equation where Celsius is represented by the variable \(x\) and Fahrenheit by \(y\), forming a line equation of the form \(y = mx + b\). Here, \(m\) is the slope indicating how many Fahrenheit degrees a Celsius degree corresponds to, and \(b\) is the y-intercept where the line crosses the y-axis.
The distinct freezing and boiling values provide us two points, namely \((0, 32)\) and \((100, 212)\), which are used to derive the linear equation. Understanding this conversion is immensely helpful in everyday life for weather readings, cooking, and scientific experiments.
Slope and Intercept
Intercept and slope are fundamental concepts in the mathematical representation of lines. The **slope**, \(m\), measures the steepness of the line, showing the rate of change between variables. In temperature conversion, this rate indicates how rapidly temperatures in Celsius convert to Fahrenheit. Calculated as \(\frac{y_2 - y_1}{x_2 - x_1}\), it becomes \(1.8\) from the points \((0, 32)\) to \((100, 212)\). This means for every 1°C increase, Fahrenheit increases by 1.8°F.
The **y-intercept**, \(b\), represents the point where the line crosses the y-axis. It indicates the Fahrenheit equivalent when Celsius is 0. In our temperature equation, it is \(32\), which means at 0°C, it is 32°F. By combining both, the equation \(y = 1.8x + 32\) is formed. This provides a clear method to transform temperatures between the scales.
The **y-intercept**, \(b\), represents the point where the line crosses the y-axis. It indicates the Fahrenheit equivalent when Celsius is 0. In our temperature equation, it is \(32\), which means at 0°C, it is 32°F. By combining both, the equation \(y = 1.8x + 32\) is formed. This provides a clear method to transform temperatures between the scales.
Mathematical Modeling
Mathematical modeling involves representing real-world phenomena through equations or data sets. In the case of temperature conversion, we accurately represent the relationship between Celsius and Fahrenheit using a linear model. This model helps predict unknown values easily, like finding the Fahrenheit equivalent for a given Celsius temperature.
By deriving an equation from known data points, like the freezing and boiling points of water, we create a reliable model \(y = 1.8x + 32\). This model is valuable as it simplifies converting any Celsius reading into Fahrenheit confidently. For example, to find out what 20°C equates to in Fahrenheit, simply substitute into the equation: \(y = 1.8 \times 20 + 32\), giving 68°F.
Through mathematical modeling, complex real-world systems become understandable and predictable, aiding in planning and decision-making.
By deriving an equation from known data points, like the freezing and boiling points of water, we create a reliable model \(y = 1.8x + 32\). This model is valuable as it simplifies converting any Celsius reading into Fahrenheit confidently. For example, to find out what 20°C equates to in Fahrenheit, simply substitute into the equation: \(y = 1.8 \times 20 + 32\), giving 68°F.
Through mathematical modeling, complex real-world systems become understandable and predictable, aiding in planning and decision-making.
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