Problem 114 - A Graphical Approach to Precalculus with Limits

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 inequal. ity that corresponds to this range.

Step-by-Step Solution

Answer

The inequality is \(|F - 730| \leq 50\).

Step 1: Identify the Range

The temperature range given is from 680°F to 780°F. This means that the values of the temperature, denoted as \( F \), must be greater than or equal to 680°F and less than or equal to 780°F.

Step 2: Find the Midpoint

To express this range as an absolute value inequality, we first calculate the midpoint of the range. The midpoint \( M \) is given by the average of the two bounds:\[ M = \frac{680 + 780}{2} = \frac{1460}{2} = 730 \]

Step 3: Determine the Range from the Midpoint

Next, find how far the endpoints of the range are from the midpoint. This is the maximum deviation from the midpoint:\[ D = 780 - 730 = 50 \]Thus, the temperature can deviate by 50°F from the midpoint 730°F.

Step 4: Write the Absolute Value Inequality

Using the midpoint and the deviation, we express the range as an absolute value inequality:\[ |F - 730| \leq 50 \]This inequality states that the temperature \( F \) is within 50°F of the midpoint 730°F, covering the range from 680°F to 780°F.

Key Concepts

Temperature Conversion

Temperature conversion is about changing a temperature measurement from one scale to another. The most common temperature scales used are Celsius (°C), Fahrenheit (°F), and Kelvin (K). For example, to convert Celsius to Fahrenheit, you use the formula:

• Multiply the Celsius temperature by \( \frac{9}{5} \).
• Add 32 to the result.

So, \[F = C \times \frac{9}{5} + 32\]For Fahrenheit to Celsius conversion, you reverse the formula:

• Subtract 32 from the Fahrenheit temperature.
• Multiply the result by \( \frac{5}{9} \).

This gives us the formula: \[C = (F - 32) \times \frac{5}{9}\]Understanding these conversion formulas is essential when you need to work with temperature data in different scales, like from scientific measurements or weather reports.

Range Finding

Range finding is about determining how much variation or spread exists within a set of data. In temperature measurements, this means identifying the minimum and maximum values. Once these limits are found, you can describe the data's span. For our example, the industrial process operates between 680°F and 780°F.
The range provides a clear picture of the acceptable operational temperatures. By setting these limits, industries can ensure safety and efficiency within their processes.
In practical terms, defining a range helps manage conditions to stay within desired limits.

• Defines safety parameters.
• Allows for planning and process optimization.
• Helps to monitor and maintain quality.

Midpoint Calculation

Midpoint calculation is a practical way to find the "center" of a range. In our exercise, the midpoint represents the central temperature: 730°F. This is very important for understanding symmetry in data.
To calculate it, simply take two numbers (the bounds of the range) and calculate their average. This can be calculated using the formula:\[M = \frac{\text{Lower bound} + \text{Upper bound}}{2}\]For example, between our temperatures, 680°F and 780°F:

• Add 680 and 780 to get 1460.
• Divide by 2 to find the midpoint: 730°F.

Finding the midpoint can simplify complex ranges and help establish central points of interest or importance. It is particularly valuable in data analysis and scientific calculations for balancing and identifying the center in diverse datasets.