Problem 30 - Algebra 2

Question

Ice forms at a temperature of \(0^{\circ} \mathrm{C},\) which corresponds to a temperature of \(32^{\circ} \mathrm{F}\). A temperature of \(100^{\circ} \mathrm{C}\) corresponds to a temperature of \(212^{\circ} \mathrm{F} .\) What temperature is the same on both scales?

Step-by-Step Solution

Answer

The temperature is -40 degrees, which is the same on both the Celsius and Fahrenheit scales.

Step 1: Understand the Relationship

To find the temperature where Celsius and Fahrenheit scales are the same, we use the conversion formula: \[ F = \frac{9}{5}C + 32 \] where \(F\) is the temperature in Fahrenheit and \(C\) is the temperature in Celsius. We need to find a temperature \(T\) where \(F = C = T\).

Step 2: Set Up the Equation

Since we want \(F = C\), let \(F = C = T\). Substitute \(T\) into the conversion formula:\[ T = \frac{9}{5}T + 32 \] This represents the mathematical condition we are trying to solve, where temperature \(T\) is the same on both scales.

Step 3: Solve for T

Rearrange the equation \[ T = \frac{9}{5}T + 32 \] to isolate \(T\):\[ T - \frac{9}{5}T = 32 \]Convert \(T\) to a common fraction form to make calculations easier:\[ \frac{5}{5}T - \frac{9}{5}T = 32 \]This simplifies to:\[ \frac{-4}{5}T = 32 \]Multiply both sides by \(-\frac{5}{4}\) to solve for \(T\):\[ T = \frac{32 \times (-5)}{4} = -40 \]Thus, \(T = -40\).

Step 4: Verify the Solution

Substitute \(T = -40\) back into the conversion formula to verify:\[ F = \frac{9}{5}(-40) + 32 = -72 + 32 = -40 \]Since both calculations agree, \(T = -40\) is the correct solution.

Key Concepts

Celsius to Fahrenheit conversion

When converting a temperature from Celsius to Fahrenheit, the aim is to express the temperature using the Fahrenheit scale. The relationship between these two scales is established by a specific formula:\[ F = \frac{9}{5}C + 32 \]where:

• \(F\) is the temperature in Fahrenheit.
• \(C\) is the temperature in Celsius.

This formula is derived from understanding the relationship between the freezing and boiling points of water in both scales:

• Water freezes at \(0^{\circ}C\) and \(32^{\circ}F\).
• Water boils at \(100^{\circ}C\) and \(212^{\circ}F\).

The conversion formula uses these key points to determine the scale conversion. Converting Celsius to Fahrenheit can be visualized as an adjustment of the Celsius value by multiplying with \(\frac{9}{5}\) and then adding \(32\) to shift the zero point from Celsius to Fahrenheit.

Fahrenheit to Celsius conversion

Knowing how to convert temperature from Fahrenheit to Celsius is just as important, especially when handling scientific measurements gathered in Fahrenheit. This conversion is done using the reverse formula:\[ C = \frac{5}{9}(F - 32) \]Here,

• \(C\) stands for degrees Celsius.
• \(F\) stands for degrees Fahrenheit.

To convert Fahrenheit into Celsius, you start by subtracting \(32\) from the Fahrenheit temperature. This adjustment is necessary because \(32^{\circ}F\) is the point where water freezes, representing \(0^{\circ}C\). Then, the result is multiplied by \(\frac{5}{9}\) to adjust the scale difference, converting the temperature effectively into Celsius units.

Equal temperatures in Celsius and Fahrenheit

Some temperatures read the same on both Celsius and Fahrenheit scales, which might seem curious at first. This intersection point occurs because each scale increases at a different rate after accounting for its zero-degree mark. This specific temperature is found using algebraic methods, setting a single variable for both scales:1. Let the temperature be \( T \).2. Use the equation \( F = T \) and \( C = T \).3. Apply the formula for conversion: \[ T = \frac{9}{5}T + 32 \]By solving the equation, as shown in the step-by-step solution, you find \( T = -40 \). This means at \(-40^{\circ}\), both Celsius and Fahrenheit scales read the same.

Temperature formulas

Understanding how temperature conversion formulas work is essential. These formulas translate temperature measurements between the two most commonly used scales worldwide. Let's break down their components:- **Celsius to Fahrenheit:** The formula used is \[ F = \frac{9}{5}C + 32 \]
It converts Celsius into Fahrenheit by scaling and shifting the Celsius temperature.- **Fahrenheit to Celsius:** The formula is \[ C = \frac{5}{9}(F - 32) \]
Here, it reverses the conversion process, undoing the Fahrenheit scaling and shifting.These formulas are rooted in the fixed points of water's freezing and boiling temperatures and help align temperature values across scales with different interval divides. Understanding and applying these can assist in grasping how the scales relate to real-world phenomena. Both formulas are practical tools for any scientific or everyday temperature conversions.