Problem 76 - Algebra A Combined Function

Question

Solve. The coldest temperature ever recorded in the United States was \(-80^{\circ} \mathrm{F}\) in Alaska. The warmest temperature ever recorded was \(134^{\circ} \mathrm{F}\) in California. How many degrees warmer is \(134^{\circ} \mathrm{F}\) than \(-80^{\circ} \mathrm{F} ?\) (Source: The World Almanac, 2005)

Step-by-Step Solution

Answer

The temperature is 214 degrees warmer.

Step 1: Understanding the Problem

We need to find the temperature difference between the coldest temperature recorded in the U.S. at \(-80^{\circ} \mathrm{F}\) and the warmest temperature at \(134^{\circ} \mathrm{F}\). This involves calculating how many degrees warmer the warmest temperature is when compared to the coldest.

Step 2: Setting Up the Calculation

To find the difference, subtract the colder temperature (\(-80^{\circ} \mathrm{F}\)) from the warmer temperature (\(134^{\circ} \mathrm{F}\)). The difference between the two temperatures will give us how many degrees warmer it is.

Step 3: Performing the Subtraction

Calculate the temperature difference: \[ \text{Difference} = 134^{\circ} \mathrm{F} - (-80^{\circ} \mathrm{F}) \]This can be rewritten as:\[ \text{Difference} = 134^{\circ} \mathrm{F} + 80^{\circ} \mathrm{F} \]Combine the temperatures to find the total difference.

Step 4: Finding the Final Answer

Add the temperatures: \[ 134 + 80 = 214 \]This means that \(134^{\circ} \mathrm{F}\) is \(214\) degrees warmer than \(-80^{\circ} \mathrm{F}\).

Key Concepts

Understanding Subtraction in Math

Subtraction is a fundamental mathematical operation that essentially means finding the difference between two numbers. Imagine a number line: moving "backwards" indicates subtraction, while moving "forwards" implies addition. In problems like temperature differences, subtraction helps to determine the variance in value between two points.

Whenever you see something like "how many more" or "how much less," subtraction is typically involved. Here’s how you approach it:

• Identify the two values you are comparing (e.g., warmest and coldest temperatures).
• Subtract one value from the other to find the difference.
• The equation can be written as: Difference = Higher Value - Lower Value.

In this exercise, we needed to calculate how much warmer 134°F is compared to -80°F. We used subtraction to determine this difference. Instead of subtracting directly, we transformed subtraction involving a negative number into addition, which simplifies the process.

Working with Negative Numbers

Negative numbers can feel a bit tricky at first, but they are quite useful in math, especially when dealing with temperature. A negative number is any number that is less than zero. Think of it as moving to the left side of zero on a number line.

In our exercise, you encounter \(-80^{\circ} \mathrm{F}\), which is a negative number.

• Negative numbers are often used to represent temperatures below freezing or other sub-zero values.
• When subtracting a negative number, you effectively add its positive counterpart. This is because subtracting a negative is the same as adding a positive.

In the temperature problem, \(134^{\circ} \mathrm{F} - (-80^{\circ} \mathrm{F})\) becomes \(134^{\circ} \mathrm{F} + 80^{\circ} \mathrm{F}\), simplifying to 214°F. This operation shows how negative numbers can be converted to simplify a calculation.

Temperature Conversion Basics

Temperature conversion allows us to switch from one temperature scale to another, such as Fahrenheit to Celsius, and vice versa. Although not directly needed in this specific exercise, understanding conversion helps when comparing temperatures internationally.

Here’s a quick guide on how to convert between common temperature scales:

• Fahrenheit to Celsius: Use the formula: \[ C = \frac{5}{9}(F - 32)\]
• Celsius to Fahrenheit: Use the formula: \[ F = \frac{9}{5}C + 32\]
• Kelvin to Celsius: \[ C = K - 273.15\]
• Celsius to Kelvin: \[ K = C + 273.15\]

These formulas are especially useful in scientific contexts or when dealing with weather data from different parts of the world. If someone asks whether converting the temperatures in our exercise would affect the outcome, it wouldn't change the relative difference, only the values on another scale.