Problem 70 - 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. \(23^{\circ} \mathrm{F}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Formula

The given formula to convert Fahrenheit to Celsius is \( c = \frac{5(F-32)}{9} \). Here, \( F \) represents the temperature in Fahrenheit, and \( c \) will be the temperature in Celsius after conversion.

Step 2: Substitute the Given Fahrenheit Value

Substitute the given value \( 23^{\circ} \mathrm{F} \) into the formula. This means replacing \( F \) with 23: \[c = \frac{5(23-32)}{9}\]

Step 3: Calculate the Difference

First, calculate the difference \( 23 - 32 \), which is the expression in the parenthesis:\[23 - 32 = -9\]

Step 4: Substitute the Difference

Substitute \( -9 \) back into the equation:\[c = \frac{5(-9)}{9}\]

Step 5: Simplify the Expression

Calculate the numerator \( 5(-9) = -45 \) and then divide by 9:\[c = \frac{-45}{9}\]

Step 6: Divide to Find Celsius

Divide \( -45 \) by 9 to find the temperature in Celsius:\[c = -5\]

Step 7: Final Answer

The temperature \( 23^{\circ} \mathrm{F} \) corresponds to \( -5^{\circ} \mathrm{C} \).

Key Concepts

Fahrenheit to Celsius formula

To convert temperatures from Fahrenheit to Celsius, we use a specific mathematical equation known as the Fahrenheit to Celsius formula. This conversion is key when moving from the American system of temperature measurement to the metric system commonly used across many other parts of the world.

The formula is expressed as:

• \( c = \frac{5(F-32)}{9} \)

Here, \( F \) represents the temperature in degrees Fahrenheit, and \( c \) will be the resulting temperature in degrees Celsius after performing the conversion. This formula allows us to understand the precise relationship between the two scales by incorporating the difference in their starting points and increments.

The number 32 in the formula arises because 32 degrees Fahrenheit is the point where water freezes. This is equivalent to 0 degrees Celsius, which is why it’s subtracted from the Fahrenheit value before converting. Multiplying by \( \frac{5}{9} \) adjusts for the different sizes of the degree increments in the two systems.

metric system

The metric system is an internationally accepted system of measurement that uses units such as meters, liters, and grams. It's known for its simplicity and ease of use, primarily because it applies consistent decimal-based units that facilitate easy conversions.

One of the significant components of the metric system is the Celsius temperature scale. This scale is favored globally due to its straightforward approach to defining temperature:

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

These defining points are chosen to simplify calculations and offer a direct, logical framework for understanding temperature changes. Within science and everyday life, using the metric system ensures easy communication and avoids confusion in temperature-related discussions or analyses.

Celsius scale

The Celsius scale is a temperature scale that forms a part of the broader metric system, used by scientists and in general communications worldwide. It’s set primarily with two fixed points: the freezing point of water at 0 degrees Celsius and the boiling point of water at 100 degrees Celsius under standard atmospheric conditions.

This scale is intuitive and convenient for many reasons:

• The 0 to 100 range divides easily into smaller increments, making it user-friendly for precise temperature readings.
• Its direct correlation with fundamental physical phenomena, like the phase changes of water, provides a visual and practical understanding of temperature.

The Celsius scale is named after the Swedish astronomer Anders Celsius, who first proposed a similar temperature system. Its adoption in the metric system has made it the primary scale used in scientific work and many parts of the world for everyday temperature measurements.