Problem 22 - Prealgebra

Question

We have mentioned the two temperature scales, Fahrenheit and Celsius. Table 1 is intended to give you a more intuitive idea of the relationship between the two temperatures scales. (TABLE CANNOT COPY) Table 2 gives the formulas, in both symbols and words, that are used to convert between the two scales. (TABLE CANNOT COPY) Let \(C=37\) in the formula \(F=\frac{9}{5} C+32,\) and solve for \(F\) Does the value of \(F\) agree with the information in Table \(1 ?\)

Step-by-Step Solution

Answer

When \( C = 37 \), \( F = 98.6 \). It should match the value from Table 1.

Step 1: Identify the Formula

We are given the formula for converting Celsius to Fahrenheit: \[ F = \frac{9}{5} C + 32 \] where \( C \) is the temperature in Celsius and \( F \) is the temperature in Fahrenheit.

Step 2: Substitute the Given Value

We need to find \( F \) when \( C = 37 \). Substitute \( C = 37 \) into the formula:\[ F = \frac{9}{5} \times 37 + 32 \]

Step 3: Multiply Fractions

Calculate \( \frac{9}{5} \times 37 \):\[ \frac{9 \times 37}{5} = \frac{333}{5} = 66.6 \]

Step 4: Add 32 to the Result

Add 32 to the product we calculated from the fraction:\[ F = 66.6 + 32 = 98.6 \]

Step 5: Check Against Table

Compare \( F = 98.6 \) with the value from Table 1. The Fahrenheit temperature should match the given information.

Key Concepts

Fahrenheit to Celsius

Understanding how to convert temperatures from Fahrenheit to Celsius is useful for science classes and daily life. People in different countries use one scale or the other, which makes this conversion important. The formula to convert a Fahrenheit temperature to a Celsius temperature is:

• \[ C = \frac{5}{9}(F - 32) \]
• Here, \( C \) is the temperature in Celsius, and \( F \) is the temperature in Fahrenheit.

The subtraction of 32 is necessary because Fahrenheit sets its freezing point of water at 32 degrees, while Celsius begins at 0 degrees. Multiplying by \( \frac{5}{9} \) scales the temperature change in Fahrenheit to one equivalent in Celsius, a scale where water freezes at 0 and boils at 100.

Celsius to Fahrenheit

Converting Celsius to Fahrenheit is essential for understanding temperatures in systems that use different scales. If you know a temperature in Celsius and need it in Fahrenheit, use this formula:

• \[ F = \frac{9}{5}C + 32 \]
• In this equation, \( C \) represents the Celsius temperature, and \( F \) is the temperature in Fahrenheit.

Begin by multiplying the Celsius temperature by \( \frac{9}{5} \). This adjusts for the larger scale range in Fahrenheit. Then add 32, which accounts for the Fahrenheit scale starting point, reflecting the freezing point of water. This simple method gives you the new temperature in Fahrenheit quickly.

Mathematical Formulas

Mathematical formulas serve as the backbone for temperature conversions between Fahrenheit and Celsius. These formulas are straightforward, allowing us to easily switch from one temperature scale to the other.

• Key Elements: Formulas are consistent. You always subtract or add the number related to the freezing point and scale the difference appropriately.
• The Formula Logic: Consider a general format: Multiply the input temperature by a specific fraction, reflecting the proportional difference between the two scales (\( \frac{9}{5} \) or \( \frac{5}{9} \)). Then, correct for initial scale differences by adding or subtracting 32.

These formulas do more than just convert; they illustrate how temperature scales differ and align theoretically.