Problem 29 - 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 corresponds to \(20^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

20°C is 68°F.

Step 1: Understand the Problem

We need to convert from Celsius to Fahrenheit. We know that 0°C is 32°F and 100°C is 212°F.

Step 2: Find the Conversion Formula

The relation between Celsius and Fahrenheit can be derived using the two points (0°C, 32°F) and (100°C, 212°F). This forms a linear equation of the form: \[F = mC + b,\]where \(m\) is the slope and \(b\) is the y-intercept.

Step 3: Calculate the Slope (m)

Using the points (0, 32) and (100, 212), the slope \(m\) can be calculated as:\[ m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8. \]

Step 4: Calculate the Y-Intercept (b)

Using the point (0, 32) in the equation \(F = mC + b\), substitute the known values to find \(b\):\[ 32 = 1.8 \times 0 + b \Rightarrow b = 32. \]

Step 5: Write the Conversion Formula

Using the calculated values of \(m\) and \(b\), the conversion formula from Celsius to Fahrenheit is:\[ F = 1.8C + 32. \]

Step 6: Convert 20°C to Fahrenheit

Substitute \(C = 20\) into the conversion formula to find the temperature in Fahrenheit:\[ F = 1.8 \times 20 + 32 = 36 + 32 = 68.\]

Key Concepts

Temperature Conversion Formula

To convert a temperature from Celsius to Fahrenheit, we use a specific formula. This formula is derived from the linear relationship between Celsius and Fahrenheit. The conversion formula is given by:\[ F = 1.8C + 32 \]where:

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

This formula helps to seamlessly convert temperatures by multiplying the Celsius value by 1.8 and then adding 32. This step adjusts for the differences in the size of the units and the offset between the two scales.

Linear Equations

Linear equations are mathematical expressions used to model straight-line relationships. They usually take the form:\[ y = mx + b \]where:

• \( y \) is the dependent variable,
• \( m \) is the slope of the line,
• \( x \) is the independent variable,
• \( b \) is the y-intercept.

In our temperature conversion problem, this equation form helps link Celsius (as the independent variable) to Fahrenheit (as the dependent variable). The calculated formula in our problem is a perfect illustration of how real-life transformations can be described using linear equations.

Slope Calculation

The slope of a line in a graph is a measure of how steep that line is. In the context of our temperature conversion, the slope indicates how much the temperature in Fahrenheit changes with a change in Celsius. The slope \( m \) is calculated by:\[ m = \frac{\Delta F}{\Delta C} = \frac{212 - 32}{100 - 0} = 1.8.\]This calculation uses two known points: (0°C, 32°F) and (100°C, 212°F). Here's how you calculate it:

• Find the difference in Fahrenheit temperatures (\(\Delta F\)).
• Find the difference in Celsius temperatures (\(\Delta C\)).
• Divide the change in Fahrenheit by the change in Celsius to get the slope.

The slope of 1.8 means that for every degree increase in Celsius, the Fahrenheit temperature increases by 1.8 degrees.

Y-intercept

The y-intercept of a line is where the line crosses the y-axis. It represents the value of \( y \) when \( x \) is 0. In our conversion formula, the y-intercept \( b \) is the temperature in Fahrenheit when the Celsius temperature is 0. By substituting \( C = 0 \) into our linear equation:\[ 32 = 1.8 \times 0 + b \]We find:\[ b = 32. \]This means that when it is 0°C, the corresponding temperature in Fahrenheit is 32°F. The y-intercept provides a starting point for the conversion line and is crucial to ensure that our formula correctly matches known temperature values.