Problem 43 - Fundamentals of Mathematics

Question

For the following 4 problems, perform the indica ted operations The low temperature today in Denver was \(-4^{\circ} \mathrm{F}\) and the high was \(-42^{\circ} \mathrm{F}\). What is the temperature difference?

Step-by-Step Solution

Answer

The temperature difference is 38°F.

Step 1: Identify the Temperatures

First, note the given temperatures: the low temperature is -4°F and the high temperature is -42°F.

Step 2: Use the Temperature Difference Formula

The formula to find the temperature difference when both temperatures are given in negative is the absolute difference. If the temperatures \(T_1\) and \(T_2\) are in negative, their difference is given by \(|T_1 - T_2|\).

Step 3: Calculate the Difference

Plug the temperatures into the formula: \[|-4 - (-42)| = |-4 + 42| = |38|\]

Step 4: Interpret the Absolute Value

The absolute value of 38 is 38, so the temperature difference between the low and high temperatures is 38°F.

Key Concepts

Absolute Value

Absolute value is a mathematical concept that refers to the distance of a number from zero on the number line, regardless of its direction. This means it is always a non-negative number.
For example, the absolute value of both 8 and -8 is 8, as both are exactly 8 units away from zero, just in different directions.
When calculating temperature differences, absolute value helps us find the positive difference, ensuring the result is always a non-negative number even if temperatures themselves are negative.

• Absolute value is denoted by vertical bars: \( |x| \).
• It simplifies expressions by removing negativity: \( |-4| = 4 \, \, |-42| = 42 \).

Understanding absolute value allows us to compare temperatures or other quantities without being misled by negative signs. It ensures the difference is a true measure of distance on the number line.

Negative Temperatures

Negative temperatures are values less than zero, commonly used in weather and science to describe conditions colder than a freezing point on a given scale.
For example, -4°F represents a temperature below zero on the Fahrenheit scale.
In mathematics, handling these values requires us to pay attention to their properties, especially when performing operations like subtraction.

• The negative sign indicates a point to the left of zero on a number line.
• Combining negative temperatures involves using addition rules, because subtracting a negative becomes an addition.

For instance, in our exercise, when finding the difference between -4°F and -42°F, we add 42 to -4, because subtracting a negative number results in addition: \(-4 - (-42) = -4 + 42\). This illustrates how understanding negative values is crucial for accurate calculations.

Mathematical Operations

Mathematical operations are essential for solving problems involving numbers, including negative temperatures and absolute values. Common operations include addition, subtraction, multiplication, and division.
In the exercise, you are tasked with finding the temperature difference, which primarily involves subtraction and the use of absolute value:

• When subtracting numbers, if both numbers are negative, it is like adding the absolute values: \(-4 - (-42) = -4 + 42\).
• Absolute value operation ensures the result is a positive difference: \(|38| = 38\).

Applying these operations accurately requires attention to rules governing negative numbers and absolute value.
This ensures our results are both mathematically correct and practically meaningful, like determining how much warmer or colder one temperature is than another.