Problem 122 - College Algebra

Question

When temperature is 0 degrees Celsius, the Fahrenheit temperature is \(32 .\) When the Celsius temperature is \(100,\) the corresponding Fahrenheit temperature is 212 . Express the Fahrenheit temperature as a linear function of \(C,\) the Celsius temperature, \(F(C)\) a. Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius. b. Find and interpret \(F(28)\). c. Find and interpret \(F(-40)\).

Step-by-Step Solution

Answer

a. The rate of change is 1.8. b. \(F(28) = 82.4\), meaning 28°C equals 82.4°F. c. \(F(-40) = -40\), indicating Celsius and Fahrenheit are equal at -40.

Step 1: Identify the Linear Function Formula

The relationship between Celsius (C) and Fahrenheit (F) can be expressed as a linear function: \( F(C) = mC + b \), where \( m \) is the slope or rate of change, and \( b \) is the y-intercept. We have two points: \((0, 32)\) and \((100, 212)\).

Step 2: Calculate the Slope

To find the slope \( m \), use the formula \(m = \frac{F_2 - F_1}{C_2 - C_1}\). Plug in the given points: \( m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \). This means for every 1 degree increase in Celsius, Fahrenheit increases by 1.8 degrees.

Step 3: Determine the Y-intercept

Since we already know that when \( C = 0 \), \( F = 32 \), we can identify the y-intercept \( b = 32 \). Hence, the equation becomes \( F(C) = 1.8C + 32 \). This forms the linear function relating Fahrenheit and Celsius temperatures.

Step 4: Interpretation of the Linear Function

Using \( F(C) = 1.8C + 32 \), we understand that for each degree increase in Celsius, the Fahrenheit temperature increases by 1.8 degrees. The y-intercept indicates that when the temperature is 0°C, it is 32°F.

Step 5: Calculate and Interpret F(28)

Substitute \( C = 28 \) into the function: \( F(28) = 1.8 \times 28 + 32 = 82.4 \). Thus, if the temperature is 28°C, the corresponding Fahrenheit temperature is 82.4°F.

Step 6: Calculate and Interpret F(-40)

Substitute \( C = -40 \) into the function: \( F(-40) = 1.8 \times (-40) + 32 = -40 \). At -40°C, the Fahrenheit temperature is also -40°F, showing that -40 is the point where Celsius and Fahrenheit scales intersect.

Key Concepts

Celsius to Fahrenheit Conversion

When we think about temperature conversion, especially between Celsius and Fahrenheit, understanding how each scales is crucial. The equation for converting Celsius to Fahrenheit is expressed as a linear function. This ensures a consistent transformation is applied across different temperatures.

• Celsius and Fahrenheit scales are commonly used in different regions.
• The relationship between these scales can be precisely defined using a linear equation: as temperature in Celsius increases, the Fahrenheit equivalent also rises accordingly.

The most straightforward way to understand this conversion is through the equation: \[ F(C) = 1.8C + 32 \] Here, replacing \( C \) with the Celsius temperature gives us the Fahrenheit temperature. By knowing two critical points - freezing point \((0, 32)\) and boiling point \((100, 212)\) - we are able to deduce this conversion function effectively.

Slope Calculation

The slope of a line indicates how steep it is, and how much one variable changes with respect to another. In the context of the temperature conversion problem, the slope \( m \) reflects how Fahrenheit changes when Celsius changes by a degree.

• This is a linear relationship, meaning it follows the format of \( y = mx + b \).
• The slope, \( m \), is calculated using two known points.

To find the slope in our Celsius to Fahrenheit equation, we use: \[ m = \frac{212 - 32}{100 - 0} = 1.8 \] This is interpreted to mean that with each unit increase in Celsius, the corresponding temperature in Fahrenheit increases by 1.8 degrees. This constant rate of change characterizes a linear relationship, making prediction and estimation straightforward.

Y-intercept

The y-intercept is a key component of linear equations, often represented as \( b \) in the formula \( y = mx + b \). For temperature conversion, it gives meaning to the value of the Fahrenheit temperature when Celsius is zero.

• The y-intercept is where the line crosses the y-axis.
• It provides a starting point or baseline value.

In the context of converting Celsius to Fahrenheit, knowing the y-intercept simplifies the conversion by acting as a base adjustment factor: At \( C = 0 \), \( F(C) = 32 \), which is our y-intercept. This shows that at the freezing point in Celsius, the Fahrenheit scale reads 32 degrees. It forms the backbone of the conversion formula, allowing us to adjust from any given Celsius temperature.

Temperature Conversion

The process of temperature conversion is about understanding how to switch between different units of measuring temperature, such as Celsius and Fahrenheit. By establishing a relationship between the two, individuals can easily determine the equivalent measure in another scale.

• Both Celsius and Fahrenheit have distinct starting points and intervals.
• Conversion allows for uniform understanding and application in varying contexts.

Using the equation \( F(C) = 1.8C + 32 \), anyone can convert a given temperature from Celsius to Fahrenheit by:1. Multiplying the Celsius temperature by 1.8.2. Adding 32 to the result.
For example, finding \( F(28) \) involves calculating \( 1.8 \times 28 + 32 = 82.4 \). Conversely, for \( F(-40) \), you'll find \( 1.8 \times (-40) + 32 = -40 \). This valuable tool not only aids daily weather reporting but also scientific applications requiring precise temperature measurement. Understanding these conversions ensures accurate translations between scales and no misinterpretation of temperature data.