Problem 57 - Algebra A Combined Function

Question

Solve. See Example 22. The lowest temperature ever recorded in Massachusetts was \(-35^{\circ} \mathrm{F}\). The highest recorded temperature in Massachusetts was \(142^{\circ}\) higher than the record low temperature. Find Massachusetts' highest recorded temperature. (Source: National Climatic Data Center)

Step-by-Step Solution

Answer

The highest recorded temperature in Massachusetts was 107°F.

Step 1: Understand the problem

We need to find the highest recorded temperature in Massachusetts. We know that this temperature is 142 degrees Fahrenheit higher than the lowest recorded temperature, which was -35°F.

Step 2: Formulate the equation

Let's denote the highest recorded temperature as \( T \). According to the problem, the highest recorded temperature is 142°F more than the lowest recorded temperature, which can be expressed as \( T = -35 + 142 \).

Step 3: Solve the equation

To find \( T \), simply add the two numbers: \( -35 + 142 = 107 \).

Step 4: Interpret the result

The calculated result, \( T = 107 \), represents the highest recorded temperature in Massachusetts in degrees Fahrenheit.

Key Concepts

Equations

Understanding equations is essential in solving algebra word problems. An equation is a mathematical statement that shows the equality between two expressions, often including variables and constants. In the given problem, the challenge is to find an unknown temperature value, and an equation can help us do that.

Here, we are trying to determine the highest recorded temperature in Massachusetts. We use equations to express relationships between known and unknown quantities. In our problem:

• The lowest temperature is known: \(-35^{\circ}\mathrm{F}\)
• The information that the highest temperature is 142°F higher than the lowest.

We represent the highest temperature as a variable, say \( T \). The equation for our problem is:

• \( T = -35 + 142 \)

By setting up this equation, we clarify the relationship and make it easy to solve for \( T \). It succinctly conveys all the necessary information and gives us a direct way to find the solution.

Temperature Conversion

In many problems about temperature, conversion between different scales (like Celsius to Fahrenheit) can be necessary. However, this particular problem only requires an understanding of temperatures in degrees Fahrenheit. There is no direct conversion needed, but grasping the arithmetic operations on temperature values is crucial.

• Temperature in Fahrenheit can be negative, which often causes confusion in calculations.
• Adding or subtracting numbers from these temperatures adjusts the scale proportionately.

When you perform operations, like adding 142 to the lowest recorded temperature of \(-35^{\circ}\mathrm{F}\), focus on the arithmetic process involved. Adding and subtracting temperatures directly relates to simple addition or subtraction.

Hence, understanding temperature arithmetic is critical to solving problems in this domain efficiently.

Problem-Solving Steps

Approaching algebra word problems often involves executing structured problem-solving steps, which bolster understanding and enhance accuracy. Let's break these steps down as they apply to the given problem:1. **Understand the Problem:** Comprehend the factual details presented. Here, the fact that the highest temperature is 142°F higher than the lowest is essential.
2. **Formulate an Equation:** Establish a variable to represent the unknown, then construct an equation from the problem’s details. For our exercise, we used:

• \( T = -35 + 142 \)

3. **Solve the Equation:** Carry out the calculations. The equation simplifies to finding \( T \) as follows:

• \( -35 + 142 = 107 \)

4. **Interpret the Results:** Link the solution back to the problem's context. The computed \( T = 107 \) indicates the highest recorded temperature. Following these problem-solving steps can significantly streamline tackling similar algebra word problems, leading to solutions that are both correct and meaningful.