Problem 28 - Prealgebra

Question

The Google Earth image is of Lake Clark National Park in Alaska. Lake Clark has an average temperature of 40 degrees Fahrenheit. What is its average temperature in Celsius to the nearest degree? (picture not copy)

Step-by-Step Solution

Answer

4 degrees Celsius.

Step 1: Understand the Conversion Formula

To convert a temperature from Fahrenheit to Celsius, we use the formula: \( C = \frac{5}{9} (F - 32) \), where \( F \) is the temperature in Fahrenheit and \( C \) is the temperature in Celsius.

Step 2: Substitute the Value

Substitute the average temperature of Lake Clark given in Fahrenheit into the formula. Here, \( F = 40 \). So, the expression becomes: \( C = \frac{5}{9} (40 - 32) \).

Step 3: Perform the Subtraction

Subtract 32 from 40: \( 40 - 32 = 8 \). So, the expression now is \( C = \frac{5}{9} \times 8 \).

Step 4: Perform the Multiplication

Multiply 8 by \( \frac{5}{9} \): \( C = \frac{5 \times 8}{9} \). Simplifying this gives \( C = \frac{40}{9} \).

Step 5: Simplify the Fraction

Calculate the division: \( \frac{40}{9} \approx 4.44 \).

Step 6: Round to the Nearest Degree

Round 4.44 to the nearest whole number to find the average temperature in Celsius: 4.

Key Concepts

Celsius Conversion Formula

Converting temperatures from Fahrenheit to Celsius is a foundational concept in understanding temperature scales. To do this, we need to use the Celsius conversion formula: \[ C = \frac{5}{9} (F - 32) \] Here, \( C \) represents the temperature in Celsius and \( F \) is the temperature in Fahrenheit. This formula comes from two important points:

• The freezing point of water is 32 degrees Fahrenheit and 0 degrees Celsius.
• The boiling point of water is 212 degrees Fahrenheit and 100 degrees Celsius.

The conversion formula essentially adjusts the Fahrenheit temperature by removing the 32-degree offset that aligns it with the Celsius freezing point, followed by scaling the result by \( \frac{5}{9} \) since there are five Celsius degrees for every nine Fahrenheit degrees. Understanding this equation helps in converting manually and comprehending the relationship between these two temperature scales.

Fahrenheit to Celsius

The process of converting a specific temperature from Fahrenheit to Celsius involves using the established formula. For instance, to convert 40 degrees Fahrenheit to Celsius, follow the outlined process:Start by substituting the Fahrenheit value into the formula. For 40°F, it becomes:\[ C = \frac{5}{9} (40 - 32) \]The basic idea is that you first adjust for the freezing point of water (32°F) by performing the subtraction. This step aligns with the Celsius scale, which starts at zero degrees for freezing water. Following the subtraction, the remaining Fahrenheit value is scaled by the fraction \( \frac{5}{9} \) to match the Celsius scale.This scaling reflects the difference in the size of the degree units between Fahrenheit and Celsius, where nine Fahrenheit degrees equal five Celsius degrees. This straightforward approach can be universally applied to any Fahrenheit to Celsius conversion.

Temperature Calculation Steps

Calculating the temperature from Fahrenheit to Celsius involves several steps that simplify the conversion process. Following the steps ensures accurate results:

• Subtraction: Start by subtracting 32 from the Fahrenheit temperature to adjust the scale (e.g., \( 40 - 32 = 8 \)).
• Multiplication: Multiply the result by \( \frac{5}{9} \) to convert to Celsius (e.g., \( C = \frac{5}{9} \times 8 \)). This represents changing from Fahrenheit degrees to Celsius units.
• Division: Carry out the division \( \frac{40}{9} \) resulting in an approximate Celsius value (e.g., 4.44).
• Rounding: To report an understandable and practical number, round the result to the nearest degree, which gives 4°C for this instance.

Each of these steps aligns with the mathematical operations needed for precise conversion. This methodical progression helps ensure each transformation is clear and reliable.