Problem 67 - 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. \(68^{\circ} \mathrm{F}\)

Step-by-Step Solution

Answer

The temperature is \( 20^{ullet}C \).

Step 1: Identify the given Fahrenheit temperature

The problem provides a temperature in degrees Fahrenheit, denoted by \( F \). Here, \( F = 68 \) degrees Fahrenheit.

Step 2: Apply the conversion formula

Use the formula to convert degrees Fahrenheit to degrees Celsius: \[c = \frac{5(F-32)}{9}\] Plug in \( F = 68 \) into the formula.

Step 3: Substitute Fahrenheit temperature into the formula

Substitute \( F = 68 \) into the formula: \[c = \frac{5(68-32)}{9}\] First perform the subtraction inside the parentheses: \( 68 - 32 = 36 \).

Step 4: Calculate the product in the numerator

Multiply the result of the subtraction by 5: \[c = \frac{5 \times 36}{9}\] This gives us \( c = \frac{180}{9} \).

Step 5: Divide to find the Celsius temperature

Finally, perform the division: \[c = \frac{180}{9} = 20\] Thus, the temperature in degrees Celsius is \( 20^{ullet}C \).

Key Concepts

Celsius Temperature

The Celsius scale is one of the most commonly used temperature scales worldwide, particularly in countries that employ the metric system. It is named after the Swedish astronomer Anders Celsius, who first proposed it in 1742. The Celsius scale is based on the properties of water, a substance crucial to life on Earth. Water freezes at 0°C and boils at 100°C under standard atmospheric pressure. This centigrade scale—which comes from the Latin word meaning 'hundred steps'—offers a simple way to measure temperature based on these two fixed points.

Celsius is ideal for scientific work and everyday use, as the intervals are intuitively structured around the behavior of water. Hence, it's often the preferred choice in most fields of study and domestic environments outside the United States, which predominantly uses the Fahrenheit scale. The Celsius notation is straightforward, typically presented as a number followed by the symbol °C, making it easy to interpret the measure of how hot or cold something is.

Temperature Conversion Formula

Converting temperatures from Fahrenheit to Celsius is essential when dealing with different measurement systems. Thankfully, there's a formula tailored for this conversion:

• The formula is \( c = \frac{5(F-32)}{9} \).
• Here, \( F \) stands for the temperature in degrees Fahrenheit.
• \( c \) represents the temperature in degrees Celsius.

This conversion formula is crafted to account for the differences in scale and offset inherent in the Fahrenheit and Celsius systems.

Let's break this formula down. First, you take the Fahrenheit temperature and subtract 32, accounting for the freezing point of water in the Fahrenheit scale. Next, you multiply the result by 5 and then divide by 9. This ratio of 5/9 scales the temperature appropriately to the Celsius scale.

Using this formula allows you to easily switch between systems, helping you understand temperatures in both domestic and scientific settings. The formula is not only practical but also crucial for anyone involved in activities requiring international cooperation or travel.

Metric System Temperature Measurement

The metric system is a globally adopted system of measurement that is simple and efficient, with the Celsius scale being its standard for temperature. Originating in the late 18th century amid growing scientific advancement, it provides a universal way of understanding measurements.

• In the metric system, base units are logical and interrelated.
• The Celsius scale is part of this system, providing a cohesive method of temperature measurement.
• Other units in the metric system, like meters for length and kilograms for mass, integrate seamlessly with the Celsius temperature scale.

Using Celsius within the metric system offers clarity and precision, especially in scientific communities. It simplifies calculations, as it's directly tied to the physical properties of water—a common reference in science.

Global sciences, engineering projects, and international trade benefit significantly from the use of Celsius. It eliminates confusion by standardizing temperature measurement, establishing a common language across borders. While the Kelvin scale might be used for more specialized scientific work, Celsius remains the everyday choice, thanks to its practicality and ease of understanding.