Problem 75 - Algebra A Combined Function

Question

Solve. The coldest temperature ever recorded on Earth was \(-129^{\circ} \mathrm{F}\) in Antarctica. The warmest temperature ever recorded was \(136^{\circ} \mathrm{F}\) in the Sahara Desert. How many degrees warmer is \(136^{\circ} \mathrm{F}\) than \(-129^{\circ} \mathrm{F}\) ? (Source: Questions Kids Ask, Grolier Limited, 1991 , and The World Almanac)

Step-by-Step Solution

Answer

The temperature difference is 265°F.

Step 1: Understand the Problem

We need to determine how many degrees warmer the warmest temperature, \(136^{\circ} \mathrm{F}\), is compared to the coldest temperature, \(-129^{\circ} \mathrm{F}\). Essentially, we are calculating the difference between these two temperatures.

Step 2: Analyze the Temperature Difference

To find the difference between two temperatures, subtract the colder temperature from the warmer one. This involves subtracting a negative number.

Step 3: Set up the Subtraction Equation

Set up the equation to find the difference: \(136^{\circ} \mathrm{F} - (-129^{\circ} \mathrm{F})\).

Step 4: Execute the Calculation

Subtracting a negative number is equivalent to adding its positive. Thus, the equation becomes: \(136 + 129\).

Step 5: Simplify the Addition

Perform the addition: \(136 + 129 = 265\). This is the temperature difference in degrees Fahrenheit.

Key Concepts

Subtraction of Integers

When dealing with real-world math problems like temperature differences, it's essential to understand the concept of subtracting integers. Subtraction can often seem tricky, but with a clear understanding, it becomes straightforward. In our problem, we need to find out how much warmer the high temperature is than the low temperature, which involves subtracting one integer from another.

Let's break down the math: suppose you have two numbers, the warmer temperature (positive integer) and the colder temperature (negative integer). To solve this, you subtract the colder temperature from the warmer temperature. This is expressed as:

• If you subtract a negative number, it's the same as adding the positive version of that number.

In our exercise, this means:

• We have to set up the equation as: \( 136^{\circ} \mathrm{F} - ( -129^{\circ} \mathrm{F}) \)

The negative sign in front of \(-129^{\circ} \mathrm{F}\) changes to a plus, making the equation: \(136 + 129\).
Understanding this rule simplifies problems involving subtraction of integers.

Absolute Value

Absolute value is a crucial mathematical concept that helps in measuring the size or magnitude of a number, regardless of its positive or negative sign. In the context of our problem, absolute value allows us to easily determine the difference between two temperatures.

Absolute value is represented by straight bars around a number, for example, \(|x|\). It represents the distance of a number from zero on the number line without considering its sign:

• For a positive number, like 136, its absolute value is \( |136| = 136 \).
• For a negative number, such as -129, its absolute value is \( |-129| = 129 \).

Using absolute value is smart when you're interested in the magnitude of a difference. By calculating the absolute values of temperatures, we find:

• The difference between \( |136| \ and \ |-129| \ is \ |136 + 129| = 265 \).

This is straightforward because absolute value removes any confusion from the negative sign, focusing purely on numeric difference.

Real World Math Problems

Real world math problems often involve scenarios where mathematical concepts must be applied to real-life situations. Understanding how to apply these concepts is key to solving them effectively. In the exercise with temperatures, we're using math to understand a difference in extremities from nature: the hottest and coldest recorded temperatures on Earth.

When applying math to real-world problems:

• Identify the real-life quantities involved (e.g., temperatures in degrees Fahrenheit).
• Understand the context (Antarctica's coldest vs. Sahara's hottest temperatures).
• Translate the situation into a mathematical formula or equation and solve it step-by-step.

In this case, calculating how much warmer it is in the Sahara Desert than in Antarctica allowed us to follow the logical steps of a math problem:

• Simplify subtraction into addition due to negative integers.
• Apply addition to find a real numerical result (265°F difference).

This method makes it easier to handle even complex math problems by breaking them down into manageable parts.