Problem 71 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve by writing a sum of signed numbers and adding. The greatest temperature variation recorded in a day is 100 degrees in Browning, Montana, on January \(23,1916\). The low temperature was \(-56^{\circ} \mathrm{F}\). What was the high temperature?

Step-by-Step Solution

Answer

The high temperature was 44 degrees Fahrenheit.

Step 1: Understand the Problem

It's given that the temperature variation is 100 degrees and the low temperature is -56 degrees. The aim is to find the high temperature. Knowing that temperature variation is computed as the difference between the high temperature and the low temperature, we can put this into an equation: High Temperature = Low Temperature + Temperature Variation.

Step 2: Insert Values into the Equation

Substitute -56 for 'Low Temperature' and 100 for 'Temperature Variation' in the above equation, So, High Temperature = -56 + 100.

Step 3: Solve the Equation

Solving the equation will give us the High Temperature. When you perform -56 + 100, it results in 44.

Step 4: Interpret the Result

The high temperature recorded that day was 44 degrees Fahrenheit.

Key Concepts

Temperature Variation

Understanding temperature variation is key to solving problems involving extreme weather changes. Temperature variation refers to the difference between the highest and lowest temperatures recorded over a specific period. In a practical example like the Browning, Montana, record, knowing the variation helps us calculate exact temperatures experienced on that day.
In this specific case, a variation of 100 degrees Fahrenheit is mentioned. The variation informs us how much the temperature climbed from a low point to reach a high point. To determine specific temperatures (like the high or low), you can use this variation along with known temperatures.

Signed Numbers

Signed numbers are fundamental in mathematics, particularly when working with temperatures. They consist of both positive and negative values, which are crucial in representing temperatures above or below zero.
For instance, in our problem, the low temperature is given as \(-56^{\circ} F\). This use of a negative sign helps signify that the temperature is below zero. When dealing with equations, it’s important to pay attention to these signs as they can change the outcome of calculations.

• Positive numbers indicate values greater than zero.
• Negative numbers indicate values less than zero.

This understanding allows us to effectively handle calculations involving signed numbers, ensuring accurate results.

Problem-Solving

Problem-solving is a systematic approach that helps address challenges or questions methodically. To solve the temperature variation problem, the following steps were used:

Firstly, understand the requirements of the problem. We knew the variation and one of the temperatures, so the goal was to find the missing temperature.

The second step was to convert the problem into a manageable equation. Recognizing that variation \(\text{= High Temperature} - \text{Low Temperature}\) simplifies the complexity of the task.

Lastly, solve the equation by plugging in known values. This strategy transforms abstract concepts into concrete numbers, making it easier to find solutions.

Equation Solving

Equation solving involves finding the unknown value in a mathematical expression. In this example, we aimed to determine the high temperature given the known temperature variation and low temperature.

To form the equation, you start with the known values and structure it to solve for the unknown:

• \(\text{High Temperature} = \text{Low Temperature} + \text{Temperature Variation}\)

Substitute the provided values into the equation, making it \(-56 + 100\). Solving this equation involves understanding how signed numbers operate: -56 is added to 100. Here, since 100 is larger, the answer is positive, resulting in 44. Hence, the high temperature was 44 degrees Fahrenheit. This straightforward process forms the basis of many algebraic solutions.