Problem 54 - Introductory and Intermediate Algebra for College Students 4th

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

a. The formula for \(C\) is \(C = \frac{5}{9}(F-32)\) \nb. The equivalent Celsius temperature for \(59^{\circ}\) Fahrenheit is approximately \(15^{\circ}\) Celsius.

Step 1: Solve for C

The assignment is to rearrange the equation \(F=\frac{9}{5}C + 32\) to solve for C: Start by subtracting 32 from both sides to get \(F-32 = \frac{9}{5}C\), then multiply both sides by \(\frac{5}{9}\) to get \(C = \frac{5}{9}(F-32)\).

Step 2: Substitute the Fahrenheit value into the equation

To find the Celsius temperature equivalent to \(59^{\circ}\) Fahrenheit, substitute \(F=59\) into the derived equation \(C = \frac{5}{9}(F-32)\). So, \(C = \frac{5}{9}(59-32)\) gives \(C \approx 15 ^{\circ}\) Celsius.

Key Concepts

Celsius to Fahrenheit

Understanding how to convert temperatures from Celsius to Fahrenheit is essential, especially when traveling between countries that use different temperature scales. The formula to convert Celsius (C) to Fahrenheit (F) is given by:
\( F = \frac{9}{5}C + 32 \)
This equation is derived from the basic principles of temperature conversion:

• Multiply the Celsius temperature by \( \frac{9}{5} \)
• Add 32 to the result of the multiplication

This conversion is based on the linear relationship between the two temperature scales. The fraction \( \frac{9}{5} \) accounts for the different size of one degree on the Celsius and Fahrenheit scales. Meanwhile, the number 32 aligns the freezing point of water at 0°C with 32°F.

Fahrenheit to Celsius

Converting Fahrenheit to Celsius is the reverse process of the previous conversion. This is useful for interpreting temperatures when moving between regions where Fahrenheit is standard. The formula used is:
\( C = \frac{5}{9}(F - 32) \)
To derive this formula, we rearrange the Celsius to Fahrenheit equation:

• First, subtract 32 from the Fahrenheit temperature to align the scales at their common melting and freezing point.
• Then, multiply the result by \( \frac{5}{9} \) to adjust for the degree size difference.

By using this formula, any Fahrenheit temperature can be efficiently converted to Celsius, thereby allowing you to better understand the temperature in Celsius context.

Solving Equations

Rearranging equations is a vital skill in mathematics, enabling you to express variables differently. For the equation \(F = \frac{9}{5}C + 32\) given in the exercise, we needed to isolate C to find its value at certain Fahrenheit temperatures. To do this:

• Subtract 32 from both sides of the equation to eliminate the constant on the right side, giving \(F - 32 = \frac{9}{5}C\).
• To remove the coefficient of \(\frac{9}{5}\), multiply both sides by the reciprocal, \(\frac{5}{9}\), resulting in \(C = \frac{5}{9}(F - 32)\).

This manipulation illustrates basic algebraic principles such as moving terms around and balancing equations. Solving for a specific variable uncovers its relationship with others, helping to accurately convert temperatures in this context.