Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The three most commonly used temperature scales are the Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\), the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\), and the Kelvin (K, an absolute temperature) scales. One interval on the Kelvin scale is equal to one degree Celsius. The freezing point of water is \(0^{\circ} \mathrm{C}\), which is \(32^{\circ} \mathrm{F}\) and \(273.15\) on the Kelvin scale. The boiling point of water is \(100^{\circ} \mathrm{C}\) and \(212^{\circ} \mathrm{F}\). On the Celsius scale, the interval between the freezing and boiling points of water is divided into 100 degrees while on the Fahrenheit scale it is divided into 180 degrees. You have been given more than enough information to answer the following questions! You ll have to select the information you will use. (a) Write a formula for a function that takes as input degrees Celsius and gives as output degrees Fahrenheit. (b) Write a formula for a function that takes as input degrees Fahrenheit and gives as output degrees Celsius. Do this as ef ciently as possible! The function you ve arrived at is the inverse of the function from part (a). (c) Write a formula for a function that takes as input degrees Celsius and gives the temperature on the Kelvin scale as output. (d) Write a formula for a function that takes as input degrees Fahrenheit and gives the temperature on the Kelvin scale as output. Express this function as the composition of two functions from previous parts of this problem.

Step-by-Step Solution

Answer

(a) \(F = \frac{9}{5}C + 32\). (b) \(C = \frac{5}{9}(F - 32)\). (c) \(K = C + 273.15\). (d) \(K = \frac{5}{9}(F - 32) + 273.15\).

Step 1: Find formula to convert Celsius to Fahrenheit

The relationship between Celsius and Fahrenheit can be described as a linear function of the form \(y = mx + b\), where \(y\) is the output in Fahrenheit, \(x\) is the input in Celsius, \(m\) is the slope, and \(b\) is the y-intercept. From the given information, we can write two equations: \[32 = m * 0 + b\] (corresponding to the freezing point), and \[212 = m * 100 + b\] (corresponding to the boiling point). Solving this system of equations gives \(m = 9/5\) and \(b = 32\). Therefore, the conversion formula is \[F = \frac{9}{5}C + 32\].

Step 2: Find formula to convert Fahrenheit to Celsius

Since the function from Celsius to Fahrenheit is a linear function, their inverse is also a linear function. Transposing the previous formula for the inverse gives: \[C = \frac{5}{9} (F - 32)\]

Step 3: Find formula to convert Celsius to Kelvin

Knowing that 1 degree change in Celsius equals a 1 degree change in Kelvin and the freezing point of water in Celsius (0°C) corresponds to 273.15K, the formula to convert Celsius to Kelvin can be determined as: \[K = C + 273.15\]

Step 4: Find formula to convert Fahrenheit to Kelvin

The conversion from Fahrenheit to Kelvin can be achieved as a composition of the two previous functions. First, convert Fahrenheit to Celsius using the formula found in step 2, then convert Celsius to Kelvin using the formula found in step 3. This leads to: \[K = \frac{5}{9} (F - 32) + 273.15\]

Key Concepts

Fahrenheit to Celsius conversion

When you're converting temperatures from Fahrenheit to Celsius, you're essentially using an inverse function of the Fahrenheit to Celsius formula. This process is important for understanding how both scales relate to each other. Let's break this down into simpler terms.

The Celsius scale is based on the properties of water, with the freezing point at 0°C and the boiling point at 100°C. The Fahrenheit scale, on the other hand, uses 32°F as the freezing point of water and 212°F as the boiling point. This difference creates a ratio that we can express as a linear function.

The conversion formula from Fahrenheit to Celsius is given by:

• Identify the difference from the freezing point: subtract 32 from the Fahrenheit value.
• Convert this difference to Celsius by multiplying by \(\frac{5}{9}\).

This results in the formula: \[C = \frac{5}{9} (F - 32)\]This equation is handy for quick conversions, especially when using a calculator. It effectively "undoes" the Fahrenheit conversion, showing how many Celsius degrees correspond to a particular Fahrenheit reading.

Celsius to Kelvin conversion

Moving from Celsius to Kelvin is probably one of the simpler conversions you'll encounter, thanks to their close relationship. The Kelvin scale is an absolute temperature scale, which means it starts at absolute zero – the coldest theoretically possible temperature – where particles have minimal motion.

However, Celsius and Kelvin scales share the same interval size, meaning one degree of temperature change is equivalent between the two. This characteristic makes the conversion straightforward as it only involves an addition of a constant.

• Take the Celsius temperature value.
• Add 273.15 to this value to shift it to the Kelvin scale.

So, the formula becomes:\[K = C + 273.15\]It's important in scientific contexts as Kelvin is often used due to its absolute nature, which makes certain types of thermodynamic calculations more consistent.

Linear functions

Understanding linear functions can greatly assist in making temperature conversions intuitive. A linear function is a mathematical way to express a direct proportionality between two quantities. It is generally written in the form \(y = mx + b\), where:

• \(y\) is the dependent variable (such as temperature in Fahrenheit).
• \(x\) is the independent variable (like temperature in Celsius).
• \(m\) is the slope, or rate of change between the two variables.
• \(b\) is the y-intercept, or the point where the line crosses the y-axis.

In the context of temperature conversion, the slope \(m\) represents how many degrees Fahrenheit change per degree Celsius, while \(b\) accounts for the offset at the freezing point of water (32°F when converting Celsius to Fahrenheit).

This means the function for converting Celsius to Fahrenheit is:\[F = \frac{9}{5}C + 32\]The slope \(\frac{9}{5}\) indicates that for every change in one degree Celsius, Fahrenheit changes by \(\frac{9}{5}\) degrees. The intercept 32 means that at 0°C, the temperature is 32°F, capturing where Celsius equals the freezing point of water on the Fahrenheit scale. By understanding this linear relationship, you gain insight into how different temperature scales measure the same physical reality of heat.