Problem 87 - Elementary and Intermediate Algebra

Question

Metallurgy. A molten alloy must be cooled slowly to control crystallization. When removed from the furnace, its temperature is \(2,700^{\circ} \mathrm{F},\) and it will be cooled at \(200^{\circ}\) per hour. Write a composition function that expresses the Celsius temperature as a function of the number of hours \(t\) since cooling began. (Hint: \(\left.C(F)=\frac{5}{9}(F-32) .\right)\)

Step-by-Step Solution

Answer

The function is \( C(t) = 1482.22 - 111.11t \).

Step 1: Establish the linear function for temperature in Fahrenheit over time

First, we need to write the function that expresses the temperature of the alloy over time in Fahrenheit. The initial temperature is \(2700^{\circ} F\) and it decreases by \(200^{\circ} F\) every hour. Thus, we use the formula \(F(t) = 2700 - 200t\), where \(t\) is the number of hours since cooling began.

Step 2: Convert the Fahrenheit function to a Celsius function

Using the conversion formula from Fahrenheit to Celsius, \(C(F) = \frac{5}{9}(F - 32)\), replace \(F\) with \(F(t)\) from the first step: \[ C(t) = \frac{5}{9}((2700 - 200t) - 32) \] Now, simplify this expression to obtain a function solely in terms of \(t\).

Step 3: Simplify the function

Expand and simplify the expression:1. Substitute \(2700 - 200t\) into the conversion formula: \[ C(t) = \frac{5}{9}(2700 - 200t - 32) \]2. Simplify inside the parentheses: \[ C(t) = \frac{5}{9}(2668 - 200t) \]3. Distribute \(\frac{5}{9}\): \[ C(t) = \frac{5}{9} \times 2668 - \frac{5}{9} \times 200t \]4. Perform the multiplication: - \(\frac{5}{9} \times 2668 \approx 1482.22\) - \(\frac{5}{9} \times 200 \approx 111.11\)Thus, the Celsius function becomes:\[ C(t) = 1482.22 - 111.11t \]

Step 4: Final Function: Write the Composition Function

The composition function that expresses the Celsius temperature \((C)\) as a function of hours \((t)\) since cooling began is:\[ C(t) = 1482.22 - 111.11t \]

Key Concepts

Temperature Conversion

Temperature conversion is a crucial concept when dealing with measurements in different units. It allows us to express temperatures in Celsius from Fahrenheit, and vice versa. Unlike objects like meters or grams, temperature scales can be quite different because they start at different points (or zero) and increase by different increments.
In our exercise, we are dealing with a conversion from Fahrenheit to Celsius. This is a common conversion because many scientific processes prefer the Celsius scale. The general formula for converting Fahrenheit (6F) to Celsius (6C) is given by:
\[C(F) = \frac{5}{9}(F - 32)\]
This formula works by first subtracting 32 from the Fahrenheit temperature, adjusting the base point to the Celsius scale. Then, it multiplies the result by \(\frac{5}{9}\) to change the scale of the increments. By understanding this formula, we can easily switch temperatures from one unit to another, as demonstrated in the exercise.

Linear Function

A linear function is essential for establishing relationships that change at a constant rate. In mathematical notation, a linear function can typically be expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In the case of the metallurgy exercise, the linear function helps us understand how the temperature of the alloy decreases over time at a consistent rate. Here, the slope of the function is -200, indicating that every hour, the temperature decreases by 200 degrees Fahrenheit. The equation for the alloy's temperature over time is represented as:

\[ F(t) = 2700 - 200t \]
In this equation, \(t\) represents time in hours, and the outcome, \(F(t)\), represents the temperature in Fahrenheit. Knowing how to set up and manipulate linear functions is a powerful tool for analyzing and predicting outcomes in various scenarios, such as the cooling of materials.

Fahrenheit to Celsius

Converting temperature readings from Fahrenheit to Celsius using a function is both practical and necessary in various scientific contexts. This is especially true since the Fahrenheit scale is mainly used in the United States, while the Celsius scale is widely used internationally.
In this exercise, we've set up a function to transform the linear cooling of an alloy from Fahrenheit to Celsius. This is done by integrating the temperature conversion formula directly into the function of temperature over time. Start by having the time-dependent function in Fahrenheit:

\[ F(t) = 2700 - 200t \]

Then, apply the conversion formula for each moment in time by replacing \(F\) with \(F(t)\):
\[ C(t) = \frac{5}{9}(F(t) - 32) \]

This transformation allows us to understand the temperature drop in coherent units (Celsius), making it fundamental for any precise analysis dependent on temperature measured in Celsius, which is more commonly used in scientific research.