Problem 54 - Introductory Algebra for College Students

Question

The formula \(F=\frac{9}{5} C+32\) expresses the relationship between Celsius temperature, \(C,\) and Fahrenheit temperature, \(F\). a. Solve the formula for \(C\). b. Use the formula from part (a) to find the equivalent Celsius temperature for a Fahrenheit temperature of \(59^{\circ}\).

Step-by-Step Solution

Answer

The equivalent Celsius temperature for a Fahrenheit temperature of \(59^{\circ}\) is \(15^{\circ} C\).

Step 1: Rearrange the formula

To rearrange the formula \(F=\frac{9}{5} C+32\) to solve for \(C\), subtract 32 from both sides to isolate the term involving \(C\) on one side and then divide by \(\frac{9}{5}\) (or multiply by \(\frac{5}{9}\)). The rearranged formula becomes \(C=\frac{5}{9}(F-32)\).

Step 2: Convert Fahrenheit to Celsius

Substitute \(F=59\) into the rearranged formula from Step 1: \(C=\frac{5}{9} (59-32)\). Perform the operations in the parentheses first, '59-32', which gives '27'. Then, multiply '27' by \(\frac{5}{9}\) to get '15'.

Key Concepts

Celsius to Fahrenheit

Understanding the conversion from Celsius to Fahrenheit is crucial when working with temperatures in different scales. The formula used to convert a temperature from Celsius (\(C\)) to Fahrenheit (\(F\)) is:

• \( F = \frac{9}{5}C + 32 \)
  

This formula essentially calculates the Fahrenheit equivalent by first multiplying the Celsius temperature by \(\frac{9}{5}\). This takes into account the difference in scale between Celsius and Fahrenheit. Since water freezes at 32°F and 0°C, we add 32 to complete the conversion.
Let's break it down further:

• Multiply the Celsius temperature by \(\frac{9}{5}\). This step converts the degree size from Celsius to Fahrenheit.
  
• Add 32 to the result of the multiplication. This accounts for the offset between the two scales (the freezing point of water).
  

By following these steps, you can effectively convert any Celsius temperature into Fahrenheit.

Fahrenheit to Celsius

The opposite conversion—changing from Fahrenheit to Celsius—is something you may need often, especially when dealing with scientific calculations. Fortunately, we can derive the formula from the Celsius to Fahrenheit equation by rearranging it:

• Solve the original formula \( F = \frac{9}{5}C + 32 \) for \( C \).
• Subtract 32 from both sides to isolate terms involving \( C \).
• Rearranging gives \( C = \frac{5}{9}(F-32) \)
  

Through this equation, you can easily convert a Fahrenheit temperature into Celsius, which is particularly useful for scientists and those living in countries using the metric system.
Here's how you can apply it:

• First, subtract 32 from the Fahrenheit temperature. This step aligns the scale with the Celsius freezing point.
  
• Then, multiply the result by \( \frac{5}{9} \). This changes the temperature units from Fahrenheit to Celsius.
  

With this method, you have a clear path from Fahrenheit to Celsius, which is part of many scientific and educational pursuits.

Rearranging Formulas

Rearranging formulas is a fundamental skill in math and science, aiding in problem-solving and better understanding of relationships between variables.
The primary goal is to isolate one of the variables, allowing you to explore different aspects of the equation. To successfully rearrange a formula like \( F = \frac{9}{5}C + 32 \) to solve for \( C \), follow these steps:

• Identify the term you need to isolate; in this case, it's \( C \).
• Perform inverse operations to both sides of the equation to move other terms away. This requires knowledge of algebraic principles, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
• Always remember to perform the same operation to each side to maintain the equation's balance.

The rearrangement can further simplify your work by providing insights into the direct influence of each variable on the result.
Practice rearranging different formulas to become more comfortable in uncovering the underlying mechanics behind math and scientific equations.