Question

The relationship between the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\) scales for measuring temperature is given by the equation $$ F=\frac{9}{5} C+32 $$ The relationship between the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and \(\mathrm{Kelvin}(\mathrm{K})\) scales is \(K=C+273 .\) Graph the equation \(F=\frac{9}{5} C+32\) using degrees Fahrenheit on the \(y\) -axis and degrees Celsius on the \(x\) -axis. Use the techniques introduced in this section to obtain the graph showing the relationship between Kelvin and Fahrenheit temperatures.

Short answer

Convert Celsius to Fahrenheit using \[F = \frac{9}{5}C + 32\]. Then convert Kelvin temperatures to Celsius to Fahrenheit. Plot the points on a graph.

Step-by-step solution

- Understand the Relationship Between Celsius and Fahrenheit

The equation given is the conversion formula from Celsius to Fahrenheit: \[F = \frac{9}{5}C + 32\] This equation will be used to graph the relationship between the two temperature scales, with Celsius on the x-axis and Fahrenheit on the y-axis.

- Understand the Relationship Between Celsius and Kelvin

The conversion formula from Celsius to Kelvin is given by: \[K = C + 273\] Kelvin temperatures can be converted back to Celsius by subtracting 273: \[C = K - 273\]

- Create the Table of Values for Celsius and Corresponding Fahrenheit

Choose a range of Celsius temperatures to calculate the corresponding Fahrenheit temperatures using the earlier formula. For example, let's take \( C = -10, 0, 10, 20, 30, 40 \). Calculate the corresponding Fahrenheit temperatures: For \( C = -10 \), \( F = \frac{9}{5}(-10) + 32 = 14 \) For \( C = 0 \), \( F = \frac{9}{5}(0) + 32 = 32 \) For \( C = 10 \), \( F = \frac{9}{5}(10) + 32 = 50 \) For \( C = 20 \), \( F = \frac{9}{5}(20) + 32 = 68 \) For \( C = 30 \), \( F = \frac{9}{5}(30) + 32 = 86 \) For \( C = 40 \), \( F = \frac{9}{5}(40) + 32 = 104 \)

- Plot the Celsius vs. Fahrenheit Data

Plot the points calculated from the table of values on a coordinate plane, with Celsius temperature on the x-axis and Fahrenheit temperature on the y-axis. For values calculated in Step 3, mark the points \((-10, 14)\), \((0, 32)\), \((10, 50)\), \((20, 68)\), \((30, 86)\), \((40, 104)\). Draw a line connecting these points to complete the graph.

- Kelvin to Fahrenheit Conversion and Interpretation

First, convert Kelvin to Celsius using the provided formula \( C = K - 273 \). Then, convert Celsius to Fahrenheit using \( F = \frac{9}{5}C + 32 \). For example:For \( K = 273 \), \( C = 273 - 273 = 0 \), \( F = 32 \).For \( K = 293 \), \( C = 293 - 273 = 20 \), \( F = 68 \).For \( K = 313 \), \( C = 313 - 273 = 40 \), \( F = 104 \).

- Plot Kelvin vs. Fahrenheit Data

Using the values calculated in Step 5, add the points \((273, 32)\), \((293, 68)\), \((313, 104)\) to the existing graph. Since the relationship between K and F is linear, connect these points to see how Kelvin temperatures translate to Fahrenheit.

Key concepts

Celsius to Fahrenheit Conversion

The relationship between Celsius \( (^{\circ}C) \) and Fahrenheit \( (^{\circ}F) \) temperatures is straightforward. The conversion formula is: \[ F = \frac{9}{5}C + 32 \] This means that to convert Celsius to Fahrenheit, you multiply the Celsius temperature by \( \frac{9}{5} \), then add 32. For instance, if you have a temperature of 25°C and want to convert it to Fahrenheit: \[ F = \frac{9}{5} \times 25 + 32 = 77^{\circ}F \] This simple formula helps with quick conversions. When plotting this relationship, Celsius values go on the x-axis, and Fahrenheit values on the y-axis.

Celsius to Kelvin Conversion

Converting Celsius \( (^{\circ}C) \) to Kelvin (K) is even simpler than converting to Fahrenheit. You just need to add 273 to the Celsius temperature: \[ K = C + 273 \] For instance, if the temperature is 25 °C, it converts to Kelvin as follows: \[ K = 25 + 273 = 298\space K \] To reverse the process, converting Kelvin back to Celsius, you simply subtract 273: \[ C = K - 273 \] For example, 300 K in Celsius would be: \[ C = 300 - 273 = 27^{\circ}C \] This linear relationship is useful for various scientific calculations.

Graphing Linear Equations

A linear equation like \[ F = \frac{9}{5}C + 32 \] represents a straight line when plotted on a graph. This line shows the relationship between Celsius and Fahrenheit temperatures. To graph this equation:

• Select a range of Celsius values (e.g., -10, 0, 10, 20, 30, 40).
• Calculate the corresponding Fahrenheit values using the formula.

For example:

• For -10°C, \[ F = \frac{9}{5}(-10) + 32 = 14^{\circ}F \]
• For 0°C, \[ F = \frac{9}{5}(0) + 32 = 32^{\circ}F \]
• For 10°C, \[ F = \frac{9}{5}(10) + 32 = 50^{\circ}F \]

Plot these points \( (-10, 14), (0, 32), (10, 50), ... \) on the coordinate system, with Celsius on the x-axis and Fahrenheit on the y-axis. Connect the dots to form a straight line, helping visualize how temperatures convert from Celsius to Fahrenheit.

Temperature Scales

Different temperature scales are used around the world for measuring temperatures. The three main scales are Celsius, Fahrenheit, and Kelvin.

• **Celsius \( (^{\circ}C) \)**: Widely used in most countries. Water freezes at 0°C and boils at 100°C.
• **Fahrenheit \( (^{\circ}F) \)**: Primarily used in the United States. Water freezes at 32°F and boils at 212°F.
• **Kelvin (K)**: Used mainly for scientific purposes. It begins at absolute zero, the lowest possible temperature. Water freezes at 273 K and boils at 373 K.

Understanding these scales' relationships is important. For example, each degree Celsius equals the same temperature change as each Kelvin. However, the Celsius and Fahrenheit scales have different starting points and increments, so knowing the conversion formulas helps in moving between these scales effectively.