Problem 96 - Precalculus Student Solutions Manual 5th

Question

When humans breathe, carbon dioxide is emitted. In one study, the emission rates of carbon dioxide by college students were measured during both lectures and exams. The average individual rate \(R_{L}\) (in grams per hour) during a lecture class satisfied the inequality $$\left|R_{L}-26.75\right| \leq 1.42,$$ whereas during an exam the rate \(R_{E}\) satisfied the inequality $$\left|R_{E}-38.75\right| \leq 2.17.$$ Use this information in Exercises. The class had 225 students. If \(T_{L}\) and \(T_{E}\) represent the total amounts of carbon dioxide in grams emitted during a one-hour lecture and exam, respectively, write inequalities that model the ranges for \(T_{L}\) and \(T_{E}\).

Step-by-Step Solution

Answer

5699.25 \leq T_{L} \leq 6338.25; 8229.50 \leq T_{E} \leq 9207.00.

Step 1: Understand the given inequalities

We have two inequalities: \[ |R_{L}-26.75| \leq 1.42 \] and \[ |R_{E}-38.75| \leq 2.17. \] These represent the range of emission rates around the average rates.

Step 2: Convert absolute inequalities to compound inequalities

The inequality \[|R_{L} - 26.75| \leq 1.42 \] can be rewritten as two inequalities: \[26.75 - 1.42 \leq R_{L} \leq 26.75 + 1.42. \] Similarly, \[|R_{E} - 38.75| \leq 2.17 \] can be rewritten as: \[38.75 - 2.17 \leq R_{E} \leq 38.75 + 2.17. \]

Step 3: Simplify the compound inequalities

Simplifying the inequalities, we get: \[25.33 \leq R_{L} \leq 28.17 \] and \[36.58 \leq R_{E} \leq 40.92. \]

Step 4: Multiply the inequalities by the number of students

To find the total amount of carbon dioxide emitted, multiply the range of rates by 225 (the number of students). For the lecture: \[225 \cdot 25.33 \leq T_{L} \leq 225 \cdot 28.17, \] and for the exam: \[225 \cdot 36.58 \leq T_{E} \leq 225 \cdot 40.92. \]

Step 5: Calculate the final ranges for total emissions

Perform the multiplication to get the range for total emissions: \[5699.25 \leq T_{L} \leq 6338.25 \] for the lecture, and \[8229.50 \leq T_{E} \leq 9207.00 \] for the exam.

Key Concepts

Compound Inequalities

Understanding compound inequalities is crucial for solving problems involving ranges and intervals. A compound inequality consists of two simple inequalities joined by either 'and' or 'or'. For example, the inequality \(|R_{L} - 26.75| \leq 1.42\) translates into \(26.75 - 1.42 \leq R_{L} \leq 26.75 + 1.42\). This represents that the rate must be within 1.42 units of the average.
Breaking an absolute value inequality into a compound inequality helps us see the range of possible values more clearly.

Carbon Dioxide Emission

Carbon dioxide emission is the release of CO2 into the atmosphere. This is a natural part of respiration in humans. In the given exercise, we consider the CO2 emissions of college students.
During lectures and exams, the emissions vary slightly, which is important for calculating overall emissions. By understanding these emissions, we can better gauge the environmental impact of human activities in confined spaces like classrooms. The exercise helps illustrate mathematical concepts using real-world data, such as understanding how a person's activity level changes their emission rate.

Multiplication of Inequalities

Multiplying inequalities involves scaling the range of values by a constant factor. In our problem, we multiply the emission rates by the number of students (225) to get total emission ranges.
This approach preserves the bounds of the inequality. For example, \(25.33 \leq R_{L} \leq 28.17\) becomes \(225 \cdot 25.33 \leq T_{L} \leq 225 \cdot 28.17\), simplifying to \(5699.25 \leq T_{L} \leq 6338.25\).
This ensures that the estimated total emissions accurately reflect individual contributions, scaled by the number of people.