Problem 39 - Algebra 2

Question

HEALTH Hypothermia and hyperthermia are similar words but have opposite meanings. Hypothermia is defined as a lowered body temperature. Hyperthermia means an extremely high body temperature. Both conditions are potentially dangerous and occur when a person's body temperature fluctuates by more than \(8^{\circ}\) from the normal body temperature of \(98.6^{\circ} \mathrm{F}\) . Write and solve an absolute value inequality to describe body temperatures that are considered potentially dangerous.

Step-by-Step Solution

Answer

Temperatures below 90.6°F or above 106.6°F are potentially dangerous.

Step 1: Define the normal range of body temperature

The normal body temperature is given as \(98.6^{\circ} \mathrm{F}\). A fluctuation of more than \(8^{\circ}\) from this value is considered dangerous. Hence, temperatures less than \(98.6^{\circ} - 8^{\circ} = 90.6^{\circ} \mathrm{F}\) or greater than \(98.6^{\circ} + 8^{\circ} = 106.6^{\circ} \mathrm{F}\) are potentially dangerous.

Step 2: Write the absolute value inequality

To capture temperatures that fluctuate more than \(8^{\circ}\) from \(98.6^{\circ} \mathrm{F}\), we set up the absolute value inequality: \[|T - 98.6| > 8\]where \(T\) represents the body temperature.

Step 3: Solve the inequality

Solving the inequality involves breaking it into two separate inequalities to capture both extremes:\[T - 98.6 > 8\]which simplifies to: \[T > 106.6\]And then, \[T - 98.6 < -8\]which simplifies to:\[T < 90.6\]Thus, the solution is \(T > 106.6\) or \(T < 90.6\).

Key Concepts

Hypothermia

Hypothermia happens when your body temperature drops below normal levels. For humans, the regular body temperature is typically around \(98.6^{\circ}\)F. If it falls too low, it can be hazardous. This dangerous condition can occur when the body temperature drops to less than \(90.6^{\circ}\)F. If you've ever been outdoors on a chilly day without proper clothing, you might have experienced mild symptoms like shivering. However, severe hypothermia can lead to serious health issues, and it's important to recognize it quickly. - Symptoms include shivering, confusion, and fatigue.- It's crucial to warm up a person with hypothermia gradually. - Use blankets, warm drinks, and move to a warmer place if possible.

Hyperthermia

Hyperthermia is the opposite of hypothermia. This condition occurs when the body temperature rises above its typical levels. In our problem, this means a body temperature higher than \(106.6^{\circ}\)F is considered extremely dangerous. High temperatures can result from hot weather or overexertion without sufficient hydration.- Symptoms to watch out for include heat stroke, extreme sweating, and dizziness.- The best way to prevent hyperthermia is to stay hydrated and take breaks in the shade if you're active on a hot day.- Immediate medical attention might be necessary if someone's temperature is rising too high.

Body Temperature Fluctuation

Our body is sensitive to temperature fluctuations. A change of more than \(8^{\circ}\) from the normal \(98.6^{\circ}\)F can have severe effects, like hypothermia or hyperthermia. These fluctuations might happen due to external conditions, like weather, or internal factors, such as illness.

• It's important to monitor body temperature during extreme weather conditions or when feeling unwell.
• Simple tools like a thermometer can help you keep track of your body temperature.
• Understanding the safe limits of body temperature helps prevent life-threatening conditions.

Solving Inequalities

Solving absolute value inequalities might sound challenging, but it becomes simpler once you break it down. In the context of body temperatures, consider the inequality \[|T - 98.6| > 8\] This represents temperatures that differ by more than \(8^{\circ}\) from the normal \(98.6^{\circ}\)F. To solve it, split the problem into two parts:1. \(T - 98.6 > 8\) which simplifies to \(T > 106.6\)2. \(T - 98.6 < -8\) which simplifies to \(T < 90.6\)Thus, temperatures are potentially dangerous if they are below \(90.6^{\circ}\)F or above \(106.6^{\circ}\)F. Understanding the logic behind solving such inequalities can help in various applications beyond health, like in physics and economics.