Question

a. Find the linear function \(C=f(F)\) that gives the reading on the Celsius temperature scale corresponding to a reading on the Fahrenheit scale. Use the facts that \(C=0\) when \(F=32\) (freezing point) and \(C=100\) when \(F=212\) (boiling point). b. At what temperature are the Celsius and Fahrenheit readings equal?

Short answer

Answer: -40°

Step-by-step solution

Find the slope (m) of the linear function

We can find the slope (m) by using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the two known points on the line. In this case, the points are \((32,0)\) and \((212,100)\): $$m = \frac{100 - 0}{212 - 32} = \frac{100}{180} = \frac{5}{9}$$

Find the y-intercept (b) of the linear function

We have the slope (m) and one point \((x_1, y_1) = (32,0)\). We'll use the point-slope form of the linear function to find the y-intercept (b): \(y - y_1 = m(x - x_1)\). Plugging in our values, we get: $$0 - 0 = \frac{5}{9}(32 - b) \Rightarrow b = 32$$

Write the linear function C=f(F)

Now that we have the slope (m) and the y-intercept (b), we can write the linear function connecting Celsius and Fahrenheit temperature scales: $$C = \frac{5}{9}(F - 32)$$

Find the temperature at which Celsius and Fahrenheit readings are equal

To find the temperature at which Celsius and Fahrenheit readings are equal, we'll set the given function equal to itself: $$C = F \Rightarrow \frac{5}{9}(F - 32) = F$$ To solve for F, we can multiply both sides by 9: $$5(F - 32) = 9F$$ Now, we'll expand and combine like terms: $$5F - 160 = 9F \Rightarrow -4F = 160$$ Divide both sides by -4 to find F: $$F = -40$$ The Celsius and Fahrenheit readings are equal at \(-40°\).

Key concepts

Temperature Conversion

Temperature conversion between Celsius and Fahrenheit is an essential concept in understanding how different systems measure temperature. Many countries use Celsius, while others like the United States, use Fahrenheit.
This discrepancy can lead to confusion, but the process of converting between these scales follows a simple linear relationship.
To convert Fahrenheit to Celsius, we use the formula \( C = \frac{5}{9}(F - 32) \). Here, \( F \) represents the Fahrenheit temperature, which we adjust by subtracting 32, then multiply by \( \frac{5}{9} \).
Conversely, to convert Celsius to Fahrenheit, we employ another straightforward formula: \( F = \frac{9}{5}C + 32 \).
Understanding these formulas ensures accurate temperature readings, useful in both scientific research and everyday life.

Celsius and Fahrenheit Scale

The Celsius and Fahrenheit scales are two different methods of measuring temperature. The Celsius scale, used worldwide and in scientific contexts, is based on the freezing and boiling points of water.
On this scale, water freezes at 0°C and boils at 100°C. In contrast, the Fahrenheit scale is commonly used in the United States.
In Fahrenheit, water freezes at 32°F and boils at 212°F.
Knowing how each scale compares in specific scenarios is crucial. For instance, room temperature is about 20°C, equivalent to 68°F. Or, if you hear the weather forecast predict a chilly 0°F, that would be about -18°C.
These examples highlight the importance of understanding both scales, especially when traveling internationally or when dealing with scientific data that uses different temperature units.

Slope Intercept Form

The slope-intercept form is a way of expressing linear functions, essential in various mathematics applications. The slope-intercept form is expressed as \( y = mx + b \), where:

• \( m \) is the slope, which indicates the steepness or incline of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Using the slope-intercept form eases the understanding and solving of linear equations.
In the context of temperature conversion, we derived the formula \( C = \frac{5}{9}(F - 32) \) using this exact form. The slope \( \frac{5}{9} \) arises from comparing the change in Celsius over the change in Fahrenheit. Likewise, the y-intercept \( -32 \) indicates the shift required to align the two temperature measurements.
Recognizing how slope and intercepts influence linear functions is critical for problem-solving in many mathematical problems beyond temperature conversions.