Problem 82 - Algebra A Combined Function

Question

Find the temperature at which the Celsius measurement and the Fahrenheit measurement are the same number.

Step-by-Step Solution

Answer

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

Step 1: Understanding the Relationship

To solve this problem, we need to understand the relationship between Celsius and Fahrenheit. The formula to convert from Celsius (\(C\)) to Fahrenheit (\(F\)) is \(F = \frac{9}{5}C + 32\). We are asked to find when these two measurements are the same, meaning when\(C = F\).

Step 2: Setting Up the Equation

We substitute the condition \(C = F\) into the conversion formula:\[ F = \frac{9}{5}C + 32 \]Which becomes:\[ C = \frac{9}{5}C + 32 \]Since \(C\) is equivalent to \(F\).

Step 3: Rearranging the Equation

Now, we need to rearrange the equation to isolate \(C\). Subtract \(\frac{9}{5}C\) from both sides of the equation:\[ C - \frac{9}{5}C = 32 \]

Step 4: Simplifying the Equation

Factor \(C\) out on the left side:\[ \left(1 - \frac{9}{5}\right)C = 32 \]which simplifies to:\[ -\frac{4}{5}C = 32 \]

Step 5: Solving for C

To find \(C\), divide both sides by \(-\frac{4}{5}\):\[ C = \frac{32}{-\frac{4}{5}} \]Thus:\[ C = 32 \times -\frac{5}{4} \]\[ C = -40 \]

Step 6: Verification

To verify, plug \(C = -40\) back into the original conversion formula:\[ F = \frac{9}{5}(-40) + 32 = -72 + 32 = -40\]Thus, \(C = -40\) confirms \(F = -40\).

Key Concepts

Understanding Temperature Scales

The concept of a temperature scale is crucial in understanding how we quantify heat and cold. Celsius and Fahrenheit are two common temperature scales used globally. The Celsius scale, also known as the centigrade scale, is primarily used internationally and is based on the freezing point (0°C) and boiling point (100°C) of water under standard atmospheric pressure. Meanwhile, the Fahrenheit scale is commonly used in the United States and assigns 32°F to the freezing point of water and 212°F to its boiling point.

The important thing to remember about temperature scales is that they represent the same physical quantity differently. This can often lead to confusion which is why having a clear understanding of conversion mechanisms is essential. When converting between these two scales, it helps to apply the conversion formula correctly to maintain accuracy.

Applying the Conversion Formula

The conversion formula is a mathematical equation that allows us to convert temperatures between Celsius and Fahrenheit. To transform a temperature from Celsius (\(C\)) to Fahrenheit (\(F\)), we use the formula:

• \[ F = \frac{9}{5}C + 32 \]

This formula shows that Celsius is multiplied by 9/5 and then we add 32 to adjust for the different starting points of the scales.

To find a common value where Celsius and Fahrenheit coincide, we set \( C = F \) in the formula. Applying this condition allows one to solve mathematically and find the temperature where both scales read the same value.

Equation Solving Techniques

Solving equations involves manipulation to isolate an unknown variable—in this case, the temperature where Celsius and Fahrenheit are equal. First, substitute the known condition into the conversion formula. When \( C = F \), the conversion equation becomes: \[ C = \frac{9}{5}C + 32 \]

Next, rearrange it by moving terms involving \( C \) to one side, resulting in \[ C - \frac{9}{5}C = 32 \]. Simplifying further, we derive \[ -\frac{4}{5}C = 32 \].

The final step in solving this equation is to divide both sides to solve for \( C \): \[ C = \frac{32}{-\frac{4}{5}} \]. This evaluates to \[ C = -40 \]. It's a straightforward yet vital skill to simplify and solve equations, showing that \(-40°C = -40°F \). Verification usually involves substituting back into the original equation to ensure the solution holds true.