Problem 69 - 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. \(41^{\circ} \mathrm{F}\)

Step-by-Step Solution

Answer

The temperature is \(5^{\circ} \mathrm{C}\).

Step 1: Understand the Formula

The formula to convert Fahrenheit (\(F\)) to Celsius (\(C\)) is given by \(c = \frac{5(F-32)}{9}\). Here, \(F\) is the temperature in Fahrenheit, and \(c\) is the corresponding temperature in Celsius.

Step 2: Substitute the Fahrenheit Value

Substitute the given Fahrenheit temperature, which is \(41^{\circ} \mathrm{F}\), into the formula: \(c = \frac{5(41-32)}{9}\).

Step 3: Simplify the Expression

First, calculate the expression within the parentheses: \(41 - 32 = 9\). Then substitute back: \(c = \frac{5 \times 9}{9}\).

Step 4: Calculate the Celsius Temperature

Perform the multiplication and division: \(c = \frac{45}{9} = 5\). Therefore, \(41^{\circ} \mathrm{F}\) is equivalent to \(5^{\circ} \mathrm{C}\).

Key Concepts

Temperature Conversion

Understanding temperature conversion is essential because it allows us to shift from one temperature scale to another. The most commonly used scales are the Celsius and Fahrenheit scales. In everyday life, especially in weather forecasts and cooking, you may come across both of these.

• Celsius Scale: Centigrade, another name for Celsius, is based on the phase changes of water. It represents the point of water freezing at 0 degrees and boiling at 100 degrees.
• Fahrenheit Scale: This is used primarily in the United States for weather and cooking temperatures. It sets water freezing at 32 degrees Fahrenheit and boiling at 212 degrees.

To convert between these scales, it is crucial to have a clear understanding of the mathematical formula involved. With clear steps, conversions become a matter of simple arithmetic, making it easy to switch between where these two scales are used.

Degrees Fahrenheit

Degrees Fahrenheit is part of a temperature scale where water boils at 212 degrees and freezes at 32 degrees. This scale was proposed by Daniel Gabriel Fahrenheit in 1724 for use with his mercury thermometer. The naming might confuse some, but here is a quick overview:

• Fahrenheit is often abbreviated as °F.
• It is widely used in the United States and some Caribbean countries.
• It provides finer gradations between freezing and boiling points compared to Celsius.

Whenever temperatures are given in degrees Fahrenheit, being able to convert these to Celsius using the appropriate formula can help integrate information and bring clarity. This is especially useful for scientists and travelers who frequently need these conversions.

Mathematical Formula

In this particular case, the mathematical formula employed is crucial to convert Fahrenheit readings to Celsius. The formula simplifies to:\[ c = \frac{5(F - 32)}{9} \]Let's dive into what each part signifies:

• The expression \((F - 32)\) adjusts the Fahrenheit reading to start at the freezing point of water, i.e., 32°F.
• Multiplying by 5 shifts from Fahrenheit increments to Celsius by scaling down the degree changes.
• Dividing by 9 does the reverse scaling from the Celsius range to the more detailed finer steps of Fahrenheit reading.

This formula effectively and quickly converts any given Fahrenheit temperature to its equivalent Celsius reading. Practicing with different values helps solidify understanding and ensures accurate calculations every time. By comprehending this conversion principle, one makes informed and accurate assessments between what seem to be divergent temperature scales.