Problem 49 - Elementary Algebra

Question

Convert the following temperatures to degrees Celsius given \(C=59(F-32),\) where F represents degrees Fahrenheit. $$ 32^{\circ} \mathrm{F} $$

Step-by-Step Solution

Answer

The temperature is 0°C.

Step 1: Identify Given Information

We are given the temperature in degrees Fahrenheit, which is \(32^{\circ} \mathrm{F}\), and need to convert it to degrees Celsius.

Step 2: Write Down the Formula for Conversion

The formula to convert Fahrenheit to Celsius is \(C = \frac{5}{9}(F - 32)\), where \(F\) represents the temperature in Fahrenheit and \(C\) is the temperature in Celsius.

Step 3: Substitute the Given Fahrenheit Value into the Formula

Substitute \(F = 32\) into the formula: \[ C = \frac{5}{9}(32 - 32) \]

Step 4: Simplify the Expression

Calculate the expression inside the parenthesis: \(32 - 32 = 0\)Now, substitute back into the formula: \[ C = \frac{5}{9} \times 0 \]

Step 5: Perform the Multiplication

Multiply the two values: \(\frac{5}{9} \times 0 = 0\)Thus, \(C = 0\).

Key Concepts

Fahrenheit to Celsius

Temperature conversion is an essential concept, especially when dealing with scientific calculations or international weather reports. When we talk about converting temperatures from Fahrenheit to Celsius, we're shifting between two of the most commonly used temperature units.
The Fahrenheit scale is predominantly used in the United States, while most other countries utilize the Celsius scale, adhering to the metric system.
This conversion is valuable because it allows one to understand temperature readings across different systems seamlessly.

• The boiling point of water is 212°F in the Fahrenheit scale and 100°C in the Celsius scale.
• The freezing point of water is 32°F in Fahrenheit, which corresponds to 0°C in Celsius.

Understanding these reference points helps when you convert everyday temperature readings.

Conversion Formula

To convert a temperature from Fahrenheit to Celsius, we use a specific conversion formula. This formula is crucial because it provides a mathematical method to switch between these two units.
The most commonly used formula is:

• \[ C = \frac{5}{9}(F - 32) \]

Where:

• \(C\) is the temperature in degrees Celsius.
• \(F\) is the temperature in degrees Fahrenheit.

This formula stems from the relationship between the fixed points on the scales—the freezing and boiling point of water—and the fact that Celsius degree intervals are smaller than Fahrenheit ones.
Understanding and memorizing this formula allows quick and accurate temperature conversions without always relying on a calculator.

Algebraic Substitution

Algebraic substitution is an essential skill in solving temperature conversion problems. It involves replacing a variable in a formula with a given value, thus simplifying the expression to find the answer.
In our problem, we start by identifying our given Fahrenheit temperature, which is 32°F as mentioned in the exercise.

• Substitute \(F = 32\) into the equation: \[ C = \frac{5}{9}(32 - 32) \]
• Calculate inside the brackets, which is the subtraction part: \[ 32 - 32 = 0 \]

By performing this subtraction step, the equation simplifies to:

• \[ C = \frac{5}{9} \times 0 \] which clearly results in \(C = 0\).

Mastering algebraic substitution helps in handling not just temperature conversions but also in solving various mathematical equations effectively and efficiently.