Problem 71 - Prealgebra

Question

In the metric system, the scale we use to measure temperature is the Celsius scale. On this scale water boils at 100 degrees and freezes at 0 degrees. When we write 100 degrees measured on the Celsius scale, we use the notation \(100^{\circ} \mathrm{C},\) which is read "100 degrees Celsius." If we know the temperature in degrees Fahrenheit, we can convert to degrees Celsius by using the formula $$c=\frac{5(F-32)}{9}$$ where \(F\) is the temperature in degrees Fahrenheit. Use this formula to find the temperature in degrees Celsius for each of the following Fahrenheit temperatures. \(14^{\circ} \mathrm{F}\)

Step-by-Step Solution

Answer

The temperature in Celsius is \(-10^{\circ} \mathrm{C}\).

Step 1: Understand the Formula

First, understand the conversion formula from Fahrenheit to Celsius: \[ c = \frac{5(F-32)}{9} \] where \(c\) represents the temperature in degrees Celsius and \(F\) represents the temperature in degrees Fahrenheit.

Step 2: Substitute the Given Temperature

We are given \(F = 14^{\circ} \mathrm{F}\). Substitute \(F = 14\) into the formula: \[ c = \frac{5(14-32)}{9} \]

Step 3: Calculate the Difference

Calculate the difference within the parentheses: \(14 - 32 = -18\). So, the equation becomes: \[ c = \frac{5(-18)}{9} \]

Step 4: Multiply the Values

Multiply the value \(-18\) by \(5\): \[ 5 \times -18 = -90 \] The equation now is \[ c = \frac{-90}{9} \]

Step 5: Divide to Find Celsius

Divide \(-90\) by \(9\): \[ c = -10 \]

Step 6: Analyze the Result

You found that \(c = -10\), which means \(14^{\circ} \mathrm{F}\) is equivalent to \(-10^{\circ} \mathrm{C}\).

Key Concepts

Temperature Conversion

Temperature conversion is a fundamental process in scientific measurements and daily life. It involves changing a temperature reading from one scale to another. This is essential because different countries and fields use different temperature scales, such as Celsius and Fahrenheit. Understanding how to convert between these scales ensures accurate communication of temperature.

The conversion is not just a direct number swap; it's a mathematical process. For Celsius and Fahrenheit, the formula used is \( c = \frac{5(F-32)}{9} \) where \( c \) is the Celsius temperature and \( F \) is the Fahrenheit temperature.

To convert Fahrenheit to Celsius:

• Subtract 32 from the Fahrenheit value.
• Multiply the result by 5.
• Then divide by 9 to get the temperature in Celsius.

Conversion tools and methods are valuable for scientists, chefs, travelers, and anyone who needs to understand temperature in different scales.

Metric System

The metric system is a decimal-based system of measurement used by most countries around the world. It features units like meters, liters, and grams, and is known for its simplicity and ease of conversion.

When it comes to temperature, the metric system uses the Celsius scale. This system is beneficial due to its logical basis where water, a common reference, boils at 100 degrees and freezes at 0 degrees Celsius.

The metric system also extends beyond temperature, providing a unified framework for all forms of measurement. This unification is efficient since each unit is based on factors of ten.

Certainly, the metric system makes scientific communication globally more precise and convenient.

Celsius Scale

The Celsius scale, sometimes called the centigrade scale, is widely used around the world for temperature measurement. Its principal features lie in its relationship to the states of water.

• Water freezes at 0 degrees Celsius
• Water boils at 100 degrees Celsius

This scale is integral to the metric system, aligning with its powers of ten, which facilitates easy calculation and comprehension. The scale was named in honor of Anders Celsius, a Swedish astronomer who first presented this temperature scale in 1742.

In scientific experiments and weather reports worldwide, degrees Celsius provides a clear and standardized temperature measure.

Fahrenheit Scale

The Fahrenheit scale is a temperature measurement system primarily used in the United States. Unlike the Celsius scale, it uses different fixed points for water and ice, which can seem less intuitive to those accustomed to the metric system.

In the Fahrenheit scale:

• Water freezes at 32 degrees Fahrenheit.
• Water boils at 212 degrees Fahrenheit.

This scale was proposed by Daniel Gabriel Fahrenheit in 1724, well known for being centered around the human experience of temperature. The Fahrenheit scale provides a larger range of whole numbers for most daily temperatures, which some people find useful.

Despite its limited geographic use today, understanding the Fahrenheit scale remains valuable given its historical importance and continued usage in regions like the U.S.