Problem 98 - A Graphical Approach to Precalculus with Limits

Question

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-43.5| \leq 8.5,\) Punta Arenas, Chile

Step-by-Step Solution

Answer

The average monthly temperature in Punta Arenas ranges from 35°F to 52°F.

Step 1: Understand the absolute value inequality

The given inequality is \(|T - 43.5| \leq 8.5\). This means that the temperature \(T\) is within 8.5 degrees of 43.5 degrees. In other words, the distance between \(T\) and 43.5 is at most 8.5.

Step 2: Remove the absolute value

To solve the inequality \(|T - 43.5| \leq 8.5\), we write two separate inequalities: 1. \(T - 43.5 \leq 8.5\) 2. \(-(T - 43.5) \leq 8.5\) or equivalently \(T - 43.5 \geq -8.5\).

Step 3: Solve the first inequality

For the inequality \(T - 43.5 \leq 8.5\), add 43.5 to both sides to solve for \(T\):\[T \leq 43.5 + 8.5 = 52\].

Step 4: Solve the second inequality

For the inequality \(T - 43.5 \geq -8.5\), add 43.5 to both sides to solve for \(T\):\[T \geq 43.5 - 8.5 = 35\].

Step 5: Combine the results

Combining the solutions from Step 3 and Step 4, we have:\[35 \leq T \leq 52\].This is the range of the average monthly temperature \(T\) for Punta Arenas, Chile.

Step 6: Interpret the result

The interpretation of the result \(35 \leq T \leq 52\) is that the average monthly temperature in Punta Arenas, Chile, varies between 35 and 52 degrees Fahrenheit.

Key Concepts

Understanding Temperature Range

The concept of a temperature range is simple but crucial in many scientific and practical applications. When we talk about the temperature range, we are referring to the span between the lowest and highest temperatures expected or observed at a given location over a specific period. In this context, the original inequality \(|T-43.5| \leq 8.5\) gives us a temperature range for Punta Arenas, Chile. It tells us that the average monthly temperature doesn't deviate more than 8.5 degrees from 43.5 degrees Fahrenheit.
So, if we solve the inequality, it gives us the specific range \(35 \leq T \leq 52\). This means that throughout the month, the temperature in Punta Arenas is expected to stay between 35 and 52 degrees Fahrenheit.
Understanding temperature ranges helps in various applications, such as in agriculture where specific temperature conditions are required for crop growth, or in designing clothing suitable for the climate.

• It helps predict weather patterns and prepare accordingly.
• Important in industries where temperature control is critical, like food storage.

Interpreting Average Monthly Temperature

The average monthly temperature is a statistical measure used to smooth out daily fluctuations and provide a clearer picture of the climate tendencies in a specific location over a month.
For Punta Arenas, the exercise indicates the average monthly temperature is centered around 43.5 degrees Fahrenheit, within a variation of 8.5 degrees. This implies that while daily temperatures might exceed or fall below these values on any given day, the calculated average for the month remains close to 43.5 degrees Fahrenheit.
Having an average monthly temperature allows meteorologists and other scientists to make informed predictions and assessments about climate trends. It's used in:

• Climate study to observe changes over time.
• Informing construction decisions, ensuring buildings can withstand typical climate conditions.
• Tourism planning, as travelers need to know general weather conditions.

Solving Absolute Value Inequalities

Absolute value inequalities can appear difficult at first, but they offer a clear visual understanding once you break them down. An absolute value inequality involves finding the set of all possible values that satisfy the condition within two bounds. In the exercise, we needed to solve the inequality \(|T-43.5| \leq 8.5\). This was broken into two separate inequalities to make solving feasible.
We addressed two scenarios: \(T - 43.5 \leq 8.5\) and \(T - 43.5 \geq -8.5\). Solving these separately gives \(T \leq 52\) and \(T \geq 35\).
Together, these solutions tell us that the value of \(T\) must lie between 35 and 52. This approach shows how absolute value problems are inherently about considering two opposite directions from a point, providing insight into range and deviation issues.

• Anyone can solve these inequalities with the right steps by "removing" the absolute value.
• Essential for solving problems involving prescribed tolerance or error margins.
• Useful in real-world applications like determining manufacturing variance tolerance.