Problem 13 - Prealgebra

Question

Solve each problem by writing and solving an equation. The temperature is \(8^{\circ} \mathrm{F}\). It is expected to fall \(5^{\circ}\) each hour for the next several hours. In how many hours will the temperature be \(-7^{\circ} \mathrm{F} ?\)

Step-by-Step Solution

Answer

It will take 3 hours for the temperature to be \(-7^{\circ} \mathrm{F}\).

Step 1: Understand the Problem

We need to find out how many hours it will take for the temperature to decrease from \(8^{\circ} \mathrm{F}\) to \(-7^{\circ} \mathrm{F}\), given that it decreases by \(5^{\circ}\) each hour.

Step 2: Set Up the Equation

We know the temperature starts at \(8^{\circ} \mathrm{F}\) and falls by \(5^{\circ}\) every hour. We can set up the equation: \(8 - 5h = -7\), where \(h\) is the number of hours.

Step 3: Solve the Equation for h

To find \(h\), we solve \(8 - 5h = -7\). Start by subtracting 8 from both sides to get \(-5h = -15\).

Step 4: Simplify to Find the Solution

Now, divide both sides by \(-5\) to isolate \(h\): \(h = 3\). This means it will take 3 hours for the temperature to fall to \(-7^{\circ} \mathrm{F}\).

Key Concepts

Understanding Temperature Changes and Its Impacts

Temperature changes are a common part of our daily climate experiences. In problem-solving scenarios, understanding how temperature fluctuates can be crucial. Here, we deal with a temperature of 8 degrees Fahrenheit, dropping steadily by 5 degrees each hour.
When the temperature decreases, it creates a cooling effect which is often encountered during nighttime or when there's a change of season. This exercise also serves as an example of forecasting temperature drop over a period of time, showing how external factors can affect temperature. By predicting such changes, we can better prepare for the effects of cool weather conditions, like deciding what clothes to wear or how to manage heating costs. Understanding this concept helps in many real-life applications.

Problem Solving with Algebraic Equations

Problem solving involving algebraic equations requires breaking down the conditions and formulating them into mathematical expressions. This helps us to conceptualize the problem and find solutions.
In the given exercise, we translate the word problem into an equation by identifying key numbers and relationships. We notice that every hour, the temperature decreases by 5 degrees. So, we express this with an equation:

• Initial: 8 degrees Fahrenheit
• Each hour decrease: -5 degrees
• Desired temperature: -7 degrees Fahrenheit

This leads to the equation: \( 8 - 5h = -7 \), where \( h \) represents the hours. Solving this gives us the hours it takes for such a decrease, and showcases the ability of algebra to model and solve real-world situations effectively.

Mastering Integer Operations

Integer operations involve performing mathematical calculations with whole numbers. Here, they help us understand how quantities change over time. The temperature change problem gives us an excellent opportunity to practice these operations, as temperatures can increase or decrease by integer amounts.
In our equation, \( 8 - 5h = -7 \), we perform the following operations:

• Subtract 8 from both sides giving: \( -5h = -15 \).
• Then, divide each side by -5 to solve for \( h \): \( h = 3 \).

This demonstrates how operations like addition, subtraction, multiplication, and division are used systematically to isolate the variable. It is essential to carefully follow these steps, as they ensure accurate results and help develop a strong foundation in math skills. Understanding integer operations makes problem-solving intuitive and straightforward, especially when dealing with everyday numeric calculations.