Problem 117 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(117-118\), use the formula \(C=\frac{5}{4}(F-32)\) to express each Fahrenheit temperature, \(F,\) as its equivalent Celsius temperature, \(C\). $$-22^{\circ} \mathrm{F}$$

Step-by-Step Solution

Answer

-22 degrees Fahrenheit is equal to -67.5 degrees Celsius.

Step 1: Identifying the given temperature

The given Fahrenheit temperature is \(-22^{\circ} \mathrm{F}\). It needs to be converted to Celsius using the given formula.

Step 2: Substituting Fahrenheit value in the formula

Substitute \(F = -22\) in the Celsius conversion formula \(C=\frac{5}{4}(F-32)\) which gives, \[C=\frac{5}{4}(-22-32)\]

Step 3: Solving the Equation

Solving the above equation, we get \[C=\frac{5}{4} * (-54)\]\[C= -67.5\]Therefore, \(-22^{\circ} \mathrm{F}\) is equal to \(-67.5^{\circ} \mathrm{C}\]

Key Concepts

Fahrenheit to Celsius Formula

The formula for converting a temperature from Fahrenheit (\textdegree F) to Celsius (\textdegree C) is crucial for scientists, chefs, travelers, and many others who need to interpret temperature data. The formula \(C = \frac{5}{9}(F-32)\) is derived from the relationship between the two temperature scales where the freezing point of water is 32\textdegree F (0\textdegree C) and the boiling point is 212\textdegree F (100\textdegree C). The fraction \(\frac{5}{9}\) represents the proportional difference in each degree on the Celsius scale when compared to the Fahrenheit scale.

Understanding how to use this formula requires recognizing the two variables involved: \(C\) and \(F\). Here \(C\) represents Celsius, and \(F\) represents Fahrenheit. When converting, you will always subtract 32 from the Fahrenheit value and then multiply the result by \(\frac{5}{9}\) to find the equivalent Celsius temperature. This equation properly aligns the two temperature scales, allowing for accurate and consistent conversions.

Substituting Values in Formulas

Substitution is a key skill when using any formula. It is the act of replacing variables with actual values. Once you have the needed values, you substitute them into the formula before solving the equation. In the context of temperature conversion, to substitute the Fahrenheit value into the conversion formula, you identify the temperature in Fahrenheit, here represented as \(F\), and replace \(F\) with the actual temperature.

For example, if you have a Fahrenheit temperature of -22\textdegree F, you would replace \(F\) with -22 in the formula \(C = \frac{5}{9}(F - 32)\) to get \(C = \frac{5}{9}(-22 - 32)\). It is crucial to maintain the order of operations, which states that subtraction inside the brackets must be completed before the multiplication outside the brackets. This ensures that the formula is implemented correctly and the resulting Celsius value is accurate.

Solving Linear Equations

Solving linear equations is a fundamental aspect of algebra that involves finding the value of the unknown variable that makes the equation true. Linear equations are typically in the form \(ax + b = 0\), where \(x\) is the unknown variable, and \(a\) and \(b\) are coefficients.

In our temperature conversion example, the equation becomes \(C = \frac{5}{9}(-54)\) after substitution. To solve this, you multiply the non-variable coefficients \(\frac{5}{9}\) by -54. This operation results in the value for \(C\), the unknown variable which, in this context, is our Celsius temperature. The ability to solve linear equations enables you to work through a variety of problems in fields as diverse as physics, economics, and everyday situations, like converting temperatures.