Problem 118 - Precalculus : Building Concepts and Connections

Question

The conversion of temperature units from degrees Fahrenheit to degrees Celsius is given by the equation \(C(x)=\frac{5}{9}(x-32),\) where \(x\) is given in degrees Fahrenheit. Let \(T(x)=70+4 x\) denote the temperature, in degrees Fahrenheit, in Phoenix, Arizona, on a typical July day, where \(x\) is the number of hours after 6 A.M. Assume the temperature model holds until 4 P.M. of the same day. Find \((C \circ T)(x)\) and explain what it represents.

Step-by-Step Solution

Answer

The composition function \((C \circ T)(x)\) is \(\frac{5}{9}*38 + \frac{20}{9}x\) which gives the temperature in Phoenix, Arizona, on a typical July day \(x\) hours after 6 A.M. in degrees Celsius.

Step 1: Understand the Composition of Functions

The composition of functions is a mathematical operation where you apply one function to the outcomes of another. In this case, the function \(T(x)\) gives us the Fahrenheit temperature at a certain time of day in Phoenix, and \(C(x)\) then converts this Fahrenheit temperature to Celsius. So, to find \((C \circ T)(x)\), we substitute \(T(x)\) into \(C(x)\).

Step 2: Substitute \(T(x)\) into \(C(x)\)

To find \(C(T(x))\), we substitute \(T(x) = 70 + 4x\) into \(C(x)\). Replace the \(x\) in \(C(x)\) with \(70 + 4x\). Hence, \(C(T(x)) = \frac{5}{9}(70+4x - 32)\).

Step 3: Simplify the equation

To simplify the equation, we will first perform the inside operation in the bracket and then multiply by \(\frac{5}{9}\). Hence, \(C(T(x))= \frac{5}{9}(38 +4x)\). Multiply each term inside the parentheses by \(\frac{5}{9}\) to get: \(C(T(x))= \frac{5}{9}*38 + \frac{20}{9}x\).

Step 4: Interpret the composition function

\((C \circ T)(x)\) means taking the temperature in Phoenix at time \(x\) (measured in hours after 6 A.M.) and converting it from degrees Fahrenheit to degrees Celsius. This mathematical model could help us understand how the temperature in Phoenix changes over time on a typical July day in a way that people more familiar with Celsius could understand.

Key Concepts

Temperature Conversion

Temperature conversion is an essential aspect of understanding and interpreting temperatures across different units. It allows us to seamlessly switch between Fahrenheit and Celsius, facilitating a broader comprehension, especially for those who are accustomed to one unit over the other.

In temperature conversion, we use mathematical equations to shift between these units. For example, visiting a country that uses a different temperature scale, understanding temperature forecasts, or discussing scientific data often requires temperature conversion for clarity:

• The Fahrenheit scale, primarily used in the United States, measures temperature where water freezes at 32°F and boils at 212°F.
• The Celsius scale, on the other hand, is widely used worldwide and is based on water freezing at 0°C and boiling at 100°C.

For seamless conversion between these two scales, we utilize the equation: \[ C(x) = \frac{5}{9}(x - 32) \]where \(x\) represents the temperature in degrees Fahrenheit, and \(C(x)\) gives us the temperature in degrees Celsius. This conversion formula is derived from the relationship between the two scales and allows us to calculate the equivalent temperature easily.

Fahrenheit to Celsius

The Fahrenheit to Celsius conversion is often crucial in many scientific and everyday scenarios. Understanding this conversion helps in interpreting data collected wherever and whenever temperatures might be measured differently.

When we need to convert temperatures from Fahrenheit to Celsius, we use a specific formula. This aids in understanding environmental, scientific, or any other data where the temperature is recorded in an unfamiliar scale. The formula we utilize is:

• Subtract 32 from the Fahrenheit temperature. This step translates the base point of the Fahrenheit scale to align with Celsius.
• Multiply the result by \(\frac{5}{9}\). This step scales down the value since the Celsius scale increment is larger than that of Fahrenheit.

As an example, if you want to convert 68°F to Celsius:\[C(x) = \frac{5}{9}(68 - 32) = \frac{5}{9}(36) = 20°C\]This conversion assists in comfortably interpreting or conversing about temperatures in parts of the world where Celsius is normative. It also helps in disciplines like science where Celsius is commonly used.

Mathematical Model

A mathematical model is a representation using mathematical concepts and language of real-world systems and scenarios. In the context of temperature conversion and function composition, these models help us predict and understand changes based on known variables and constants.

In our exercise, we explored a mathematical model composed of two functions. First, \( T(x) = 70 + 4x \) describes how the Fahrenheit temperature changes over time in Phoenix, starting from 6 A.M. This represents how the temperature increases at a constant rate, capturing the typical climate pattern in the region.

Next, by applying the function \( C(x) = \frac{5}{9}(x - 32) \), we convert these predicted Fahrenheit temperatures into Celsius, showcasing the flexibility and usefulness of mathematical models beyond mere computation:

• This model allows for easy interaction with temperature predictions in Celsius, simplifying global communications.
• It demonstrates how theoretical predictions can shift to practical applications, helping individuals plan their day or interpret foreign weather predictions.

Mathematical models like these are fundamental in various fields, providing insights into predictable patterns and aiding in strategic planning by converting abstract numbers into meaningful information.