Problem 117 - College Algebra with Modeling and Visualization

Question

Temperature Scales The Celsius and Fahrenheit scales are related by the equation \(C=\frac{5}{9}(F-32)\) These scales have the same temperature reading at a unique value where \(F=C .\) Find this temperature.

Step-by-Step Solution

Answer

The temperature where Celsius and Fahrenheit are equal is -40 degrees.

Step 1: Identify the Given Equation

The relationship between the Celsius and Fahrenheit scales is given by the equation \(C = \frac{5}{9}(F - 32)\). We need to find the temperature where these scales are equal, meaning \(F = C\).

Step 2: Set Up the Equation for Equal Temperatures

Since we need to find the temperature where both scales are equal, we set \(C = F\) in the equation. This gives us \(C = \frac{5}{9}(C - 32)\).

Step 3: Clear the Fraction by Multiplying

To eliminate the fraction, multiply both sides of the equation by 9 which gives us \(9C = 5(C - 32)\).

Step 4: Distribute and Simplify the Equation

Distribute the 5 on the right-hand side: \(9C = 5C - 160\). Then, subtract \(5C\) from both sides to get \(4C = -160\).

Step 5: Solve for C

Divide both sides of the equation by 4 to solve for \(C\): \(C = \frac{-160}{4}\), which simplifies to \(C = -40\).

Step 6: Verify the Solution

Since we set \(F = C\), and \(C = -40\), substitute \(F = -40\) into \(C = \frac{5}{9}(F - 32)\) to check. \(C = \frac{5}{9}(-40 - 32) = \frac{5}{9}(-72) = -40\), which confirms our solution.

Key Concepts

Temperature Scales

Temperature scales are crucial for understanding how we measure the warmth or coldness of an environment. The most commonly used scales in the world are Celsius and Fahrenheit.

The Celsius scale, also known as centigrade, is the standard metric system scale and is widely used in science and most of the world outside of the United States. It is based around the properties of water, with 0 degrees representing the freezing point and 100 degrees the boiling point.

The Fahrenheit scale, used mainly in the United States, also has its unique points. Here, the freezing point of water is 32 degrees, and the boiling point is 212 degrees. While they are different, these scales offer a way to measure temperature effectively in various contexts. Understanding how they relate to each other is key to solving conversion problems.

Celsius and Fahrenheit Conversion

Converting temperatures between Celsius and Fahrenheit involves a specific mathematical formula. This formula is:

• For converting Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \)
• For converting Fahrenheit to Celsius: \( C = \frac{5}{9}(F - 32) \)

These formulas help translate between the two scales depending on whether you're starting with a Celsius or Fahrenheit temperature.

Understanding these conversions is essential, especially when working in international settings or scientific research where different scales might be used. Notice that the formula for Fahrenheit to Celsius uses subtraction and a fraction to bring down the value in line with the colder nature of the Celsius readings.

Equality of Temperatures

There is only one point at which the Celsius and Fahrenheit scales read an equal temperature. This value can be found mathematically by setting the conversion formulas equal to each other.

In the problem provided, we set the equation \(C = \frac{5}{9}(F - 32)\) to find at which temperature these two readings are the same. By solving that, we discovered that both scales read -40 degrees at the same time. This is a fascinating equality, as it shows the precise point of convergence between these two distinct measurement systems.

To see the mathematics in action, remember to solve the equation by equating \(F = C\), leading us through arithmetic calculations to find \(C = -40\), illustrating how different heat measurement approaches can align. This understanding is vital for learners who need to convert temperatures accurately.