Problem 31 - Algebra A Combined Function

Question

Convert Nome, Alaska's \(14^{\circ} \mathrm{F}\) high temperature to Celsius.

Step-by-Step Solution

Answer

The temperature in Nome, Alaska is -10°C.

Step 1: Understand the Conversion Formula

To convert a temperature from Fahrenheit to Celsius, we use the formula:\[ C = \frac{5}{9} \times (F - 32) \] where \( C \) is the temperature in degrees Celsius and \( F \) is the temperature in degrees Fahrenheit.

Step 2: Substitute the Fahrenheit Temperature

Substitute the given Fahrenheit temperature \( 14^{\circ}F \) into the conversion formula:\[ C = \frac{5}{9} \times (14 - 32) \]

Step 3: Calculate the Difference

Calculate the difference between \( 14 \) and \( 32 \):\[ 14 - 32 = -18 \]

Step 4: Multiply by the Conversion Factor

Multiply the result from Step 3 by \( \frac{5}{9} \):\[ C = \frac{5}{9} \times (-18) \] which simplifies to \( C = \frac{-90}{9} \).

Step 5: Compute the Final Celsius Temperature

Divide \(-90\) by \(9\) to get the temperature in Celsius:\[ C = -10 \]

Key Concepts

Celsius and Fahrenheit

When discussing temperature, two of the most commonly used scales are Celsius and Fahrenheit. In most parts of the world, Celsius is the standard unit of temperature. However, places like the United States commonly use Fahrenheit. This is why it's useful to know how to convert between the two scales.

The Celsius scale, also known as centigrade, is based on the properties of water. It sets the freezing point of water at 0°C and the boiling point at 100°C under standard atmospheric conditions. This scale is easy to understand and is used worldwide in scientific and educational contexts.

On the other hand, the Fahrenheit scale sets the freezing point of water at 32°F and the boiling point at 212°F. This scale was originally developed by Daniel Gabriel Fahrenheit in the 18th century and is primarily used in some English-speaking countries. Understanding both scales involves recognizing their different starting points and units, which is crucial especially when interpreting weather forecasts or conducting scientific experiments.

conversion formula

The conversion formula is a mathematical equation used to translate temperatures from one scale to another. For converting Fahrenheit to Celsius, the formula is:

• \( C = \frac{5}{9} \times (F - 32) \)

where:

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

This formula takes into account the different starting points of the two scales and their unit differences. The formula subtracts 32 because 32°F is the freezing point of water on the Fahrenheit scale. The fraction \( \frac{5}{9} \) scales the difference appropriately, converting Fahrenheit degrees into Celsius degrees.

Using the conversion formula is simple. Just plug in the value you have, perform the necessary calculations, and you will receive the corresponding temperature on the Celsius scale. Understanding this formula is key to converting these values quickly and accurately.

temperature calculation

Calculating temperatures with the conversion formula involves a simple series of mathematical operations. Let's examine how you can use it in practice by converting a temperature of 14°F to Celsius.

First, place the Fahrenheit temperature into the formula:

• \( C = \frac{5}{9} \times (14 - 32) \)

Next, perform the subtraction:

• \( 14 - 32 = -18 \)

This calculation gives you a negative number because 14°F is below freezing. After obtaining this difference, multiply it by the conversion factor \( \frac{5}{9} \):

• \( C = \frac{5}{9} \times (-18) \)
• This simplifies to \( C = -10 \).

Finally, your equation shows that 14°F is equivalent to -10°C. By breaking it down, you confirm the accuracy of this negative Celsius value which corresponds closely to standard winter temperatures seen in places like Nome, Alaska.