Problem 38 - Algebra for College Students

Question

(a) The equation \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\) can be used to convert from degrees Celsius to degrees Fahrenheit. Complete the following table. \begin{tabular}{l|llllllllll} \(\mathrm{C}\) & 0 & 5 & 10 & 15 & 20 & \(-5\) & \(-10\) & \(-15\) & \(-20\) & \(-25\) \\ \hline \(\mathrm{F}\) & & & & & & & & & & \end{tabular} (b) Graph the equation \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\). (c) Use your graph from part (b) to approximate values for \(\mathrm{F}\) when \(\mathrm{C}=25^{\circ}, 30^{\circ},-30^{\circ}\), and \(-40^{\circ}\). (d) Check the accuracy of your readings from the graph in part (c) by using the equation \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\).

Step-by-Step Solution

Answer

The table is completed, the graphing shows the equation as a line, and calculations confirm graph accuracy.

Step 1: Calculate Fahrenheit for Given Celsius Values

Use the formula \( \mathrm{F} = \frac{9}{5} \mathrm{C} + 32 \) to calculate each Fahrenheit value for the Celsius values in the table. 1. For \( \mathrm{C} = 0 \), \( \mathrm{F} = \frac{9}{5} \times 0 + 32 = 32 \)2. For \( \mathrm{C} = 5 \), \( \mathrm{F} = \frac{9}{5} \times 5 + 32 = 41 \)3. For \( \mathrm{C} = 10 \), \( \mathrm{F} = \frac{9}{5} \times 10 + 32 = 50 \)4. For \( \mathrm{C} = 15 \), \( \mathrm{F} = \frac{9}{5} \times 15 + 32 = 59 \)5. For \( \mathrm{C} = 20 \), \( \mathrm{F} = \frac{9}{5} \times 20 + 32 = 68 \)6. For \( \mathrm{C} = -5 \), \( \mathrm{F} = \frac{9}{5} \times (-5) + 32 = 23 \)7. For \( \mathrm{C} = -10 \), \( \mathrm{F} = \frac{9}{5} \times (-10) + 32 = 14 \)8. For \( \mathrm{C} = -15 \), \( \mathrm{F} = \frac{9}{5} \times (-15) + 32 = 5 \)9. For \( \mathrm{C} = -20 \), \( \mathrm{F} = \frac{9}{5} \times (-20) + 32 = -4 \)10. For \( \mathrm{C} = -25 \), \( \mathrm{F} = \frac{9}{5} \times (-25) + 32 = -13 \).

Step 2: Fill in the Table with Calculated Fahrenheit Values

Insert the calculated Fahrenheit values into the corresponding spaces in the table:\[\begin{tabular}{l|llllllllll}\mathrm{C} & 0 & 5 & 10 & 15 & 20 & -5 & -10 & -15 & -20 & -25 \\hline\mathrm{F} & 32 & 41 & 50 & 59 & 68 & 23 & 14 & 5 & -4 & -13 \\end{tabular}\]

Step 3: Graph the Equation F = (9/5)C + 32

Plot the given Celsius (C) and Fahrenheit (F) values on a graph to visualize the linear relationship. The x-axis represents Celsius, and the y-axis represents Fahrenheit. Connect the points to form a straight line, which represents the equation \( \mathrm{F} = \frac{9}{5} \mathrm{C} + 32 \).

Step 4: Approximate F from Graph for C = 25, 30, -30, and -40

Use the graph from Step 3 to approximate Fahrenheit values for the given Celsius values:- For \( C = 25 \), the graph suggests \( F \approx 77 \).- For \( C = 30 \), the graph suggests \( F \approx 86 \).- For \( C = -30 \), the graph suggests \( F \approx -22 \).- For \( C = -40 \), the graph suggests \( F \approx -40 \).

Step 5: Verify Graph Readings Using the Equation

Compute the Fahrenheit values for the given Celsius values using the formula \( \mathrm{F} = \frac{9}{5} \mathrm{C} + 32 \) to check the accuracy of the graph:- For \( C = 25 \), \( \mathrm{F} = \frac{9}{5} \times 25 + 32 = 77 \).- For \( C = 30 \), \( \mathrm{F} = \frac{9}{5} \times 30 + 32 = 86 \).- For \( C = -30 \), \( \mathrm{F} = \frac{9}{5} \times (-30) + 32 = -22 \).- For \( C = -40 \), \( \mathrm{F} = \frac{9}{5} \times (-40) + 32 = -40 \).The calculated values confirm the approximations from the graph.

Key Concepts

Temperature Conversion

Temperature conversion is a common task, especially when traveling between countries that use different temperature scales. To convert temperatures from degrees Celsius (°C) to degrees Fahrenheit (°F), we use the formula:

• \( F = \frac{9}{5} C + 32 \)

This formula transforms the scale of Celsius, which is based on the freezing (0°C) and boiling points (100°C) of water, into the Fahrenheit scale. The Fahrenheit scale sets these points at 32°F and 212°F respectively.
When using this formula:

• Multiply the Celsius temperature by 9/5.
• Add 32 to the result.

For example, to convert 25°C to Fahrenheit:

• Calculate \( \frac{9}{5} \times 25 = 45 \)
• Add 32 to get 77°F.

Whether for science, travel, or day-to-day weather forecasting, understanding this conversion can be very practical.

Graphing Equations

Graphing equations is an essential skill in mathematics that provides a visual representation of the relationship between variables. For linear equations, like \( F = \frac{9}{5} C + 32 \), graphing shows how two variables interact.
To begin graphing the equation:

• Assign one axis for each variable – typically the x-axis for Celsius (C) and the y-axis for Fahrenheit (F).
• Plot calculated pairs from your conversion table as points on the graph. For example, (0, 32), (5, 41).
• Connect these points to form a straight line, which reflects the linear nature of the equation.

The line's slope, or steepness, is influenced by the 9/5 ratio in the equation, showing the consistent increase in Fahrenheit for every unit increase in Celsius. This visualization is helpful not just for converting temperatures but for understanding linear relationships in broader contexts.

Equation Verification

Verifying solutions in mathematics ensures the accuracy and reliability of results, particularly when using graphical methods. Using our temperature conversion example, you can check if your graphical approximations are correct by using the formula.
Given several Celsius values, such as 25°C, 30°C, -30°C, and -40°C:

• Calculate Fahrenheit using the formula \( F = \frac{9}{5} C + 32 \).
• For 25°C, the formula gives \(\frac{9}{5} \times 25 + 32 = 77\) confirming the graph's approximation.
• Apply this process to each value to ensure consistency between your graph and calculations.

Equation verification anchors understanding and builds confidence in mathematical practices. Though the graph provides a visual, the verified calculations ensure numerical precision.

Unit Conversion in Mathematics

Unit conversion is a crucial aspect of mathematics as it allows for the comparison and understanding of various metrics through standardized values. This extends beyond temperature to include length, time, volume, and more.
Understanding unit conversion:

• Recognizes the need to transform units to utilize particular equations and formulas correctly.
• Allows for meaningful comparisons, such as knowing how many liters are in gallons or what miles translate to in kilometers.
• You often multiply by a conversion factor, like 2.54 to convert inches to centimeters.

In the context of temperature conversion, the need arises because degrees Celsius and Fahrenheit are used differently worldwide, impacting weather forecasting, scientific research, and travel.
Building familiarity with conversion factors and rules across various measurement units enhances problem-solving capabilities and comprehension of real-world scenarios.