Problem 68 - 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. \(59^{\circ} \mathrm{F}\)

Step-by-Step Solution

Answer

The temperature is 15°C.

Step 1: Understand the formula

We need to use the given conversion formula from Fahrenheit to Celsius: \( c = \frac{5(F-32)}{9} \), where \( F \) represents the temperature in degrees Fahrenheit and \( c \) will be the temperature in degrees Celsius.

Step 2: Substitute the given Fahrenheit value

Starting with \( F = 59^{\circ} \mathrm{F} \), substitute this value into the formula: \[c = \frac{5(59 - 32)}{9}\]

Step 3: Calculate the difference inside the parentheses

Subtract 32 from 59:\[59 - 32 = 27\]

Step 4: Multiply by the factor of 5

Next, multiply the result from Step 3 by 5:\[5 \times 27 = 135\]

Step 5: Divide by 9 to complete the conversion

Finally, divide the product by 9 to find the Celsius temperature:\[c = \frac{135}{9} = 15\]

Step 6: Conclusion

Thus, the conversion from \( 59^{\circ} \mathrm{F} \) to Celsius gives us \( 15^{\circ} \mathrm{C} \).

Key Concepts

Metric System

The metric system is a system of measurement that is used worldwide. It is known for its simplicity and utility in scientific and everyday contexts. One of the most significant features of the metric system is its use of standard units like meters for length, kilograms for mass, and, importantly for this discussion, Celsius for temperature. Unlike other measurement systems, the metric system is based on powers of ten, making calculations more straightforward.
For temperature, the Celsius scale is primarily used, making it integral to the metric system. This system ensures easy conversion and understanding, as its operations are simple arthimetic. This ease of calculation is why many educational systems and scientific communities prefer and encourage the use of the metric system.

Celsius Scale

The Celsius scale is a vital component of the metric system and is used to express temperature. This scale is straightforward and intuitive for several reasons:

• The freezing point of water is defined at 0°C.
• Water boils at 100°C under standard atmospheric conditions.

These reference points provide a clear understanding of the scale, as the difference between the two is exactly 100 units, facilitating precise and easy interpretations.
The Celsius scale is also advantageous because it aligns with the Kelvin scale, which scientists use when dealing with thermodynamic temperature. Kelvin and Celsius share the same incremental degree size, making conversions simple. As you become familiar with Celsius, you'll find it convenient for both everyday use and in scientific regards.

Fahrenheit to Celsius Formula

The formula for converting Fahrenheit to Celsius is essential in adapting to metric system standards, especially when dealing with temperatures reported in Fahrenheit, which is still prevalent in some countries like the United States. The formula is:\[ c = \frac{5(F-32)}{9} \]where:

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

This equation arises from aligning the freezing and boiling points of water on the two scales and involves:

• Subtracting 32 from the Fahrenheit temperature to adjust the offset in zero points.
• Multiplying by 5, as the Celsius degree is larger than the Fahrenheit degree (specifically 5/9 the size).
• Then, dividing by 9 to finalize the conversion.

By understanding and using this formula, you can accurately convert temperatures between these units, helping bridge the gap between different measurement systems, and aligning everyday instincts with scientific standards.