Problem 90 - A Graphical Approach to College Algebra

Question

Solve each problem. When a model kite was flown in crosswinds in tests, it attained speeds of 98 to 148 feet per second in winds of 16 to 26 feet per second. Using \(x\) as the variable in each case, write absolute value inequalities that correspond to these ranges.

Step-by-Step Solution

Answer

The kite's speed inequality is \(|x - 123| \\leq 25\) and the wind's speed inequality is \(|x - 21| \\leq 5\).

Step 1: Define the variable for kite speed

The problem states that the speed of the kite varies between 98 and 148 feet per second. Let \( x \) represent the speed of the kite in feet per second. We need to express this range using an absolute value inequality.

Step 2: Write the absolute value inequality for kite speed

The speed of the kite is between 98 and 148 feet per second. This can be expressed as an absolute value inequality: \[ |x - 123| \leq 25 \] Here, 123 is the midpoint of 98 and 148, calculated as \((98 + 148) / 2\), and 25 is the distance from the midpoint to either boundary, calculated as \(148 - 123\) or \(123 - 98\).

Step 3: Define the variable for wind speed

Similarly, the problem states the wind speed varies between 16 and 26 feet per second. Let \( x \) now represent the wind speed in feet per second.

Step 4: Write the absolute value inequality for wind speed

The wind speed is between 16 and 26 feet per second. This can be expressed as an absolute value inequality: \[ |x - 21| \leq 5 \] Here, 21 is the midpoint of 16 and 26, calculated as \((16 + 26) / 2\), and 5 is the distance from the midpoint to either boundary, calculated as \(26 - 21\) or \(21 - 16\).

Key Concepts

Kite Speed and Absolute Value Inequalities

Understanding kite speed, especially in varying wind conditions, involves grasping how these speeds can be represented mathematically. When a model kite flies, its speed can range dramatically due to changing factors, such as wind. In our exercise, we are told that the speed of the kite fluctuates between 98 and 148 feet per second. To express this range using absolute value inequalities, we first identify the midpoint of these two boundaries. The midpoint is calculated as \((98 + 148) / 2 = 123\). This becomes a central value to which other speeds compare.

The distance from this midpoint to either boundary determines the range, which is equal here to 25 feet per second (i.e., \(148 - 123\) or \(123 - 98\)). We can now express this as an absolute value inequality:

• \(|x - 123| \leq 25\)

This inequality states that the kite speed \(x\) is at most 25 feet per second away from 123 feet per second, covering the whole range from 98 to 148 feet per second.

Wind Speed and Absolute Value Inequalities

Wind speed is another crucial factor affecting kite performance. In the problem, the wind speed varies between 16 and 26 feet per second, much narrower than the kite speed range. Just like with the kite speed, we find the midpoint for the wind speed, which is

• \((16 + 26) / 2 = 21\)

This midpoint helps us write an absolute value inequality that captures the range as seen in the kite speed example.

We then calculate the distance from the midpoint to each data boundary:

• \(26 - 21\) or \(21 - 16\), both equal to 5 feet per second.

Therefore, the absolute value inequality for wind speed becomes:

• \(|x - 21| \leq 5\)

This inequality ensures that the wind speed \(x\) is within 5 feet per second of 21, maintaining the necessary conditions between 16 to 26 feet per second.

Midpoint Calculations in Absolute Value Inequalities

Midpoint calculations play a vital role when dealing with absolute value inequalities. They serve as a central reference point from which deviations are measured. Here is how to find the midpoint for any given range, using the exercises above as examples.

For kite speed, which ranges from 98 to 148 feet per second, we calculate:

• \((98 + 148) / 2 = 123\)

This midpoint of 123 helps us express the speed range in absolute value form, centered around this value. Similarly, for wind speed, we find:

• \((16 + 26) / 2 = 21\)

The result of these calculations provides a foundation for establishing how far you can deviate to either side, ensuring all possible values still fall within the defined constraints. Knowing how to find and use midpoints is essential when forming expressions like \(|x - 123| \leq 25\) and \(|x - 21| \leq 5\), thus efficiently capturing the range of speeds within a specific parameter. This knowledge is crucial when working with data sets and absolute value inequalities as it allows for robust modeling of variable ranges.