Problem 87 - 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 Celsius temperature as a function of time is \(C(t) = 1482.22 - 111.11t\).

Step 1: Understand the problem

We need to find a function that gives the temperature in Celsius as a function of time in hours, starting from a given temperature in Fahrenheit and a specific cooling rate.

Step 2: Express Fahrenheit temperature as a function of time

The initial temperature of the alloy is 2700°F, and it cools at 200° per hour. Thus, the temperature after \(t\) hours is given by:\[F(t) = 2700 - 200t\]

Step 3: Convert Fahrenheit temperature to Celsius

We need to convert \(F(t)\) to Celsius using the conversion formula \(C(F) = \frac{5}{9}(F - 32)\). Substitute \(F(t)\) into the conversion formula:\[C(t) = \frac{5}{9}(2700 - 200t - 32)\]

Step 4: Simplify the Celsius expression

Simplify the expression inside the Celsius conversion:\[C(t) = \frac{5}{9}(2668 - 200t)\]Distribute the \(\frac{5}{9}\):\[C(t) = \frac{5}{9} \times 2668 - \frac{5}{9} \times 200t\]Calculate the constants:\[C(t) = \frac{5 \times 2668}{9} - \frac{5 \times 200}{9}t\]\[C(t) \approx 1482.22 - 111.11t\]

Step 5: Write the final composition function

Combine the results to get the final temperature function in Celsius:\[C(t) = 1482.22 - 111.11t\]

Key Concepts

The Role of Metallurgy in Material Cooling

Metallurgy is the science and study of metals. It’s a vast field that deals with both the chemical and physical properties of metals. One important aspect of metallurgy is understanding how alloys, which are mixtures of metals, are cooled.

When a molten alloy is being cooled, careful control of the process is crucial. This is because the rate of cooling affects the alloy’s crystalline structure – that’s its arrangement of atoms. By managing this cooling process, metallurgists can control the characteristics of the metal, such as strength, hardness, and toughness.

• Quick cooling might lead to a hard but brittle metal.
• Slow cooling could result in a softer and more ductile metal.

This ability to control metal properties is beneficial for crafting materials with specific qualities, whether that’s flexible wiring or sturdy beams. Understanding metallurgy allows engineers to create better products that cater to diverse applications, from electronics to construction.

Understanding Temperature Conversion

Temperature conversion plays a fundamental role in many scientific and engineering processes. In our context of metallurgy, we often need to translate temperatures between different scales. Here, we are focused on converting Fahrenheit to Celsius.

The formula for converting a temperature from Fahrenheit to Celsius is:

• \(C(F) = \frac{5}{9}(F - 32)\)

It is essential to comprehend this formula. It tells us that to convert Fahrenheit (F) to Celsius (C), we subtract 32 from the Fahrenheit value, which accounts for the differences in the two scales’ starting points. We then multiply by \(\frac{5}{9}\), which is the ratio of the two temperature scale intervals.

Using conversion formulas accurately is important in various practical scenarios. In our exercise, this conversion is integral for expressing the cooling temperature in a universally understandable unit, allowing for precise control and study of the cooling alloy.

Cooling Rate and Its Importance

The cooling rate of a material is a key factor in metallurgy. It is defined as the speed at which a metal loses heat overtime. This rate can significantly influence the properties of an alloy.

• For example, a slower cooling rate allows larger crystals to form, offering toughness.
• Conversely, a faster cooling rate might lead to a more rigid, yet brittle structure, due to finer grain formation.

In the given problem, the alloy cools at 200°F per hour. Understanding this rate helps us predict the material's behavior and properties after treatment.

To manage and experiment with various cooling rates, metallurgists design methods to manipulate the environment and process parameters. This could involve using different types of cooling equipment or varying the ambient conditions.
Such control ensures that the desired material characteristics are achieved, making understanding cooling rates vital for fabricating quality metal products.