Problem 88 - A Graphical Approach to College Algebra

Question

Solve each problem. The industrial process that is used to convert methanol to gasoline is carried out at a temperature range of \(680^{\circ} \mathrm{F}\) to \(780^{\circ} \mathrm{F}\). Using \(F\) as the variable, write an absolute value inequality that corresponds to this range.

Step-by-Step Solution

Answer

\(|F - 730| \leq 50\)

Step 1: Understand the Problem

The problem requires writing an inequality using absolute value to express the temperature range for converting methanol to gasoline, which is between \(680^{\circ} \mathrm{F}\) and \(780^{\circ} \mathrm{F}\).

Step 2: Calculate the Midpoint

The midpoint of the temperature range is the average of the lower and upper limits. Calculate it as follows: \[ \text{Midpoint} = \frac{680 + 780}{2} = 730^{\circ} \mathrm{F}. \] This midpoint will be the central value in our absolute value inequality.

Step 3: Calculate the Range

The range, or tolerance, is half the difference between the upper and lower limits. Calculate it as follows: \[ \text{Range} = \frac{780 - 680}{2} = 50. \] This value will be used as the tolerance in the inequality.

Step 4: Write Absolute Value Inequality

Using the midpoint and the range, write the absolute value inequality: \[ |F - 730| \leq 50. \] This inequality represents all the temperatures \(F\) that are within 50 units of the midpoint \(730^{\circ} \mathrm{F}\), covering the entire range from \(680^{\circ} \mathrm{F}\) to \(780^{\circ} \mathrm{F}\).

Key Concepts

Temperature Conversion

Temperature conversion is an important concept, especially when dealing with processes that require precise temperature control. Understanding how to convert temperatures from Fahrenheit to Celsius, or vice versa, can be critical in many industrial applications. The general formulas used for temperature conversion are:

• To convert Fahrenheit to Celsius: \( C = \frac{5}{9}(F - 32) \)
• To convert Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \)

Converting temperatures ensures that procedures maintain consistency across various systems and units. It’s especially useful in international collaboration situations, where different regions might use different units. Furthermore, understanding these conversions can help in comparing historical data or when working with equipment calibrated in different units.

Industrial Process

An industrial process refers to the systematic procedures used to convert raw materials into finished goods. In the context of methanol to gasoline conversion, specific conditions such as temperature are crucial. This process involves complex chemical reactions where methanol, a type of alcohol, is converted into hydrocarbons found in gasoline.
Industrial processes often require:

• Precise conditions like temperature and pressure to optimize efficiency.
• Controlled environments to ensure safety and quality.
• Automation and monitoring for consistency in output.

By controlling these factors, industrial processes can produce large quantities of products efficiently and safely. In the case of methanol to gasoline conversion, maintaining a temperature range between 680°F and 780°F ensures the process operates effectively.

Methanol to Gasoline

Turning methanol into gasoline is a fascinating example of chemical engineering at work. This process is significant as it offers a way to produce fuel from alternative sources, potentially reducing reliance on traditional crude oil.
The methanol-to-gasoline (MTG) process involves several stages:

• Dehydration, where methanol is converted to dimethyl ether.
• Oligomerization, involving the combination of smaller molecules to form larger, gasoline-like hydrocarbons.
• Methanation and refining to enhance quality and performance.

These steps emphasize the need for controlled temperature conditions to yield optimal results. By adhering to a specific temperature range, like 680°F to 780°F, the desired chemical reactions are facilitated, ensuring that the final product meets quality standards for use as fuel in vehicles.

Midpoint Calculation

Calculating the midpoint is a straightforward yet critical mathematical operation, especially when dealing with ranges in data like temperature. The midpoint provides a central value which is equidistant from both endpoints of a range. This helps in understanding the centermost point of a set of values.
To find a midpoint:

• Add the lower limit and the upper limit of the range together.
• Divide the sum by 2.

So for a temperature range from 680°F to 780°F, the midpoint would be \( \frac{680 + 780}{2} = 730 \)°F. This midpoint is useful in forming absolute value inequalities, as it acts as a reference around which the range is balanced, indicating that all values within 50 degrees of it fall within the specified temperature range.