Problem 116 - A Graphical Approach to Precalculus with Limits

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

Kite speed: \(|x - 123| \leq 25\); Wind speed: \(|x - 21| \leq 5\).

Step 1: Identify the Range of Speeds for the Kite

The kite's speed was tested from 98 feet per second to 148 feet per second. This range can be expressed as an inequality: \[98 \leq \text{speed of the kite} \leq 148\] where the speed of the kite is denoted by \(x.\)\(x\) represents the kite's speed in feet per second.

Step 2: Identify the Range of Wind Speeds

The crosswind speeds during the test ranged from 16 feet per second to 26 feet per second. This can similarly be written as an inequality:\[16 \leq \text{speed of the wind} \leq 26\] where the speed of the wind is denoted by \(x.\)\(x\) represents the wind speed in feet per second.

Step 3: Write Absolute Value Inequality for Kite Speed

To express the range of kite speeds using an absolute value inequality, find the midpoint of 98 and 148:\[\text{Midpoint of kite speed range} = \frac{98 + 148}{2} = 123\]Then calculate the distance from the midpoint to one end of the range:\[\text{Distance } = 148 - 123 = 25\]The absolute value inequality is:\[|x - 123| \leq 25\]

Step 4: Write Absolute Value Inequality for Wind Speed

To express the range of wind speeds using an absolute value inequality, find the midpoint of 16 and 26:\[\text{Midpoint of wind speed range} = \frac{16 + 26}{2} = 21\]Then calculate the distance from the midpoint to one end of the range:\[\text{Distance } = 26 - 21 = 5\]The absolute value inequality is:\[|x - 21| \leq 5\]

Key Concepts

Kite Speed Range

When discussing the range of kite speeds, we're focusing on the interval from 98 feet per second to 148 feet per second. This means that during the tests, the kite moved within this speed range. In mathematics, we express this interval as an inequality:

• For any speed \( x \) that the kite might achieve, it must hold true that \( 98 \leq x \leq 148 \).

By using these boundaries, we can determine all the possible speeds the kite flew at during the test.

• To express this in terms of an absolute value inequality, we first find the midpoint: \( \frac{98 + 148}{2} = 123 \).
• Then, the maximum variation from this midpoint is \( 25 \) (since \( 148 - 123 = 25 \)).

Thus, we can write the inequality as \(|x - 123| \leq 25\). This means that all kite speeds are within 25 units of 123 feet per second, giving us a more structured way of understanding the kite's speed range.

Wind Speed Range

Turning to the range of wind speeds, we consider the interval from 16 feet per second to 26 feet per second. During testing, the wind moved within these speeds. By writing this as an inequality, it is expressed as:

• For any wind speed \( x \) during the tests, it must satisfy \( 16 \leq x \leq 26 \).

To convert this into an absolute value inequality, similar steps as above are applied:

• First, find the midpoint of the wind speed range: \( \frac{16 + 26}{2} = 21 \).
• Then, calculate the maximum variation from this midpoint, which is \( 5 \) (since \( 26 - 21 = 5 \)).

So, the absolute value inequality is \(|x - 21| \leq 5\). This format shows that wind speeds are not more than 5 units away from 21 feet per second, providing clarity in understanding the wind conditions during the test.

Mathematical Modeling

Mathematical modeling involves using mathematical languages and symbols to describe real-world phenomena. Here, we used inequalities to model the speed ranges of both the kite and the wind. Let's break down why this modeling is helpful:

• **Real-world representation:** By using equations and inequalities, we translate real-world scenarios (like speed ranges) into mathematical expressions.
• **Simplification:** Absolute value inequalities help simplify complex scenarios, making them easier to analyze.
• **Predictive power:** Such models can predict the performance of objects like kites in slightly different conditions, based on the known range.
• **Decision-making:** Understanding these models can help in making informed decisions about kite flight and design based on the predicted outcomes under varying conditions.

Mathematical modeling serves as a bridge between theoretical math and practical applications, showing how math can help us understand and predict real things in nature.