Problem 75 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve by writing a sum of signed numbers and adding. The temperature at 8: 00 A.M. was \(-7^{\circ} \mathrm{F}\). By noon it had risen \(15^{\circ} \mathrm{F},\) but by \(4: 00 \mathrm{P} .\). \(\mathrm{M} .\) it had fallen \(5^{\circ} \mathrm{F}\). What was the temperature at 4: 00 P.M.?

Step-by-Step Solution

Answer

The temperature at 4:00 P.M. was \(3^{\circ} \mathrm{F}\).

Step 1: Identify Initial Temperature

The initial temperature given is \(-7^{\circ} \mathrm{F}\). This is the temperature at 8:00 A.M.

Step 2: Identify Temperature change by noon

The temperature increased by \(15^{\circ} \mathrm{F}\) by noon. When temperature rises, we add this change to the initial temperature. This gives us the temperature at noon.

Step 3: Calculate Temperature at noon

To calculate the temperature at noon, we add the initial temperature to the increase in temperature. This should be \(-7^{\circ} \mathrm{F}\) (initial temperature) + \(15^{\circ} \mathrm{F}\) (increase in temperature) = \(8^{\circ} \mathrm{F}\).

Step 4: Identify Temperature change by 4:00 P.M.

By 4:00 P.M., the temperature fell by \(5^{\circ} \mathrm{F}\). When temperature falls, we subtract this change from the current temperature.

Step 5: Calculate Temperature at 4:00 P.M.

To calculate the temperature at 4:00 P.M., we subtract the decrease in temperature from the temperature at noon. This should be \(8^{\circ} \mathrm{F}\) (temperature at noon) - \(5^{\circ} \mathrm{F}\) (decrease in temperature) = \(3^{\circ} \mathrm{F}\). This is the temperature at 4:00 P.M.

Key Concepts

Addition of Signed Numbers

Adding signed numbers is a fundamental skill in algebra. It involves understanding how to manage both positive and negative numbers. When you add numbers with different signs, you subtract the smaller number from the larger number and keep the sign of the larger. For instance, if you start with \(-7\, \text{°F}\) and add \(15\, \text{°F}\), you find the difference, which is \(8\, \text{°F}\), and keep the positive sign because 15 is greater than 7.
Conversely, when both numbers are negative or positive, you simply add their absolute values and keep the common sign. Understanding this concept is crucial when solving real-world problems that involve changes, such as temperature fluctuations or financial calculations.
Let's look at an example to solidify this:

• Start with \(-7\, \text{°F}\)
• Add \(15\, \text{°F}\) (increase)
• Result: \(8\, \text{°F}\)

This demonstrates how signed number addition is applied in practical scenarios.

Temperature Change

Temperature changes are a prevalent example of how signed numbers operate in real life. Such changes often involve increases (positive change) and decreases (negative change).
Consider a day that starts at \(-7\, \text{°F}\):

• The temperature rises by \(15\, \text{°F}\), leading us to add this value, resulting in \(8\, \text{°F}\) by noon.
• Later, it drops by \(5\, \text{°F}\). This drop means subtracting, resulting in \(3\, \text{°F}\) by 4:00 P.M.

Understanding how to calculate these temperature changes using signed numbers is crucial when predictions or historical data are needed. It illustrates how mathematics can apply to everyday life, offering solutions and explanations for natural phenomena.

Algebraic Reasoning

Algebraic reasoning involves using algebraic concepts to solve problems. In the context of temperature change, it's important to identify the starting point, track each change, and apply operations accordingly. This requires problem-solving skills and critical thinking.
Consider this step-by-step approach:

• Identify the initial temperature (starting point).
• Note every temperature change in sequence.
• Use addition or subtraction to calculate new temperatures.

This logical and structured approach helps students develop efficient problem-solving techniques. By managing signed numbers and understanding temperature changes, students enhance their ability to reason algebraically, preparing them for more advanced mathematical concepts.