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Chapter 2: Problem 76

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 Fanrenheit 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 \(F(-40)\) .

Short Answer

Expert verified
Rate of change is 1.8; \( F(28) = 82.4 \); \( F(-40) = -40 \), where Celsius and Fahrenheit are equal.

Step by step solution

01

Understanding the Relationship Between Celsius and Fahrenheit

To find the linear function that represents Fahrenheit as a function of Celsius, we need to understand that for a linear relationship, we can write it in the form of \[ F(C) = mC + b \]where \( m \) is the rate of change or slope, and \( b \) is the y-intercept.
02

Calculate the Rate of Change (Slope)

The rate of change is calculated as the change in Fahrenheit temperature divided by the change in Celsius temperature. Using the given points \((0, 32)\) and \((100, 212)\), we can find the slope:\[ m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \]Therefore, the rate of change is 1.8. This means for every 1-degree increase in Celsius temperature, the Fahrenheit temperature increases by 1.8 degrees.
03

Determine the Linear Function

With the slope \( m = 1.8 \) and the point \((0, 32)\), we can find the y-intercept \( b = 32 \). Thus, the linear function is:\[ F(C) = 1.8C + 32 \]
04

Calculate and Interpret F(28)

To find \( F(28) \), substitute 28 for \( C \) in the linear function:\[ F(28) = 1.8 \times 28 + 32 = 50.4 + 32 = 82.4 \]This means when the Celsius temperature is 28 degrees, the Fahrenheit temperature is 82.4 degrees.
05

Calculate and Interpret F(-40)

To find \( F(-40) \), substitute -40 for \( C \) in the linear function:\[ F(-40) = 1.8 \times (-40) + 32 = -72 + 32 = -40 \]This means when the Celsius temperature is -40 degrees, the Fahrenheit temperature is also -40 degrees. This is the point where the Celsius and Fahrenheit scales intersect (i.e., -40 degrees Celsius is equal to -40 degrees Fahrenheit).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
In mathematics, the slope refers to the rate of change between two variables. Imagine you are looking at how two types of temperatures relate: Celsius and Fahrenheit. The slope shows us how much one variable changes when the other one changes just a little. To calculate the slope for temperature conversion from Celsius to Fahrenheit, we use two known points:- Point 1: When Celsius is 0, Fahrenheit is 32.- Point 2: When Celsius is 100, Fahrenheit is 212.The formula for finding the slope (m) is:\[ m = \frac{\text{Change in Fahrenheit}}{\text{Change in Celsius}} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \]This calculation shows that for every 1 degree increase in Celsius, Fahrenheit increases by 1.8 degrees. It's like saying a gentle hill has a slightly steep climb.
Temperature Conversion
Temperature conversion involves changing temperature measurements from one unit to another, and the most common conversion is between Celsius and Fahrenheit. These units are related by a linear equation derived from the concept of slope.Here鈥檚 the general form of a linear equation that fits this conversion:\[ F(C) = mC + b \]- **m** is the slope we calculated earlier, which tells us how much Fahrenheit changes for a change in Celsius. Here, it's 1.8.- **b** is the y-intercept, which shows where our line crosses the Fahrenheit axis. For Celsius 0, it is 32.So, the formula for converting Celsius to Fahrenheit becomes:\[ F(C) = 1.8C + 32 \]This line helps us quickly figure out what temperature it is in Fahrenheit for any Celsius value.
Celsius to Fahrenheit
The practical application of converting Celsius to Fahrenheit is everywhere, especially when traveling or following recipes. To do this with our derived formula, simply substitute the Celsius value into the equation.- For example, if you want to know what 28 degrees Celsius is in Fahrenheit, use:\[ F(28) = 1.8 \times 28 + 32 = 82.4 \]- Similarly, for -40 degrees Celsius, plug the value into the equation:\[ F(-40) = 1.8 \times (-40) + 32 = -40 \]Notice how -40 degrees is a special point where the Celsius and Fahrenheit scales read the same, showcasing an interesting crossover of scales.Armed with this formula, you can easily make these conversions and see how temperature behaves across different measuring systems.

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Most popular questions from this chapter

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