Problem 58 - Beginning and Intermediate Algebra

Question

Solve. See Example 8 On January \(2,1943,\) the temperature was \(-4^{\circ}\) at 7: 30 a.m. in Spearfish, South Dakota. Incredibly, it got \(49^{\circ}\) warmer in the next 2 minutes. To what temperature did it rise by \(7: 32 ?\)

Step-by-Step Solution

Answer

The temperature rose to \(45^{\circ}\) by 7:32 a.m.

Step 1: Identify the Initial Temperature

The initial temperature at 7:30 a.m. was given as \(-4^{\circ}\). Make a note of this as it represents our starting point in calculating the new temperature.

Step 2: Determine the Temperature Change

We are informed that the temperature increased by \(49^{\circ}\) in the next 2 minutes. This is the amount the initial temperature will change.

Step 3: Calculate the New Temperature

To find the new temperature at 7:32 a.m., we add the temperature change from Step 2 to the initial temperature from Step 1: \[-4^{\circ} + 49^{\circ} = 45^{\circ}\].

Step 4: Write the Conclusion

By performing the addition, we found that the temperature at 7:32 a.m. was \(45^{\circ}\).

Key Concepts

Temperature Change in Spearfish

Temperature change is a fascinating concept that plays a significant role in many real-world scenarios, such as climate monitoring and weather forecasting. In this specific scenario from 1943, understanding the change in temperature over a short period gives us a glimpse into the dynamics of weather events. We were initially given a temperature of \(-4\degree\), serving as our baseline for calculation.

When temperatures fluctuate, they exemplify temperature change, which is essentially the difference in temperature from one point in time to another. In Spearfish, a remarkable increase of \(49\degree\) within just two minutes indicated the region's weather was highly variable.

It's important to think about temperature change as a straightforward way to measure how quickly environmental conditions can vary. Recognizing these changes aids in assessing the implications of weather patterns that can impact daily activities and broader climatic trends.

Understanding Integer Addition

Adding integers is a fundamental concept in mathematics and is particularly useful in solving real-world problems, such as calculating temperature changes. Let's break down the process of integer addition as it applies to this scenario. We start with the initial temperature of \(-4\degree\), which is an integer representing a lower temperature.

To calculate the new temperature, we perform integer addition with the amount the temperature increased by. The formula applied is simply \(-4 + 49\). Here, \(-4\) is our starting point, and we add \(49\), the increased temperature. This addition results in \(45\).

• Negative numbers represent temperatures below zero and are often initially less intuitive.
• Integer addition helps in seamlessly transitioning between negative and positive values such as temperatures rising or lowering.

Integer addition is not only about finding the sum but also offers an efficient way to handle and compute operations that involve changes in state, like temperature shifts.

Real-World Application of Temperature Calculations

Temperature calculations, like the one we just explored, demonstrate an important real-world application of mathematical concepts. These calculations allow us to comprehend and predict weather changes which impact numerous aspects of life, from agriculture to daily activities.

In Spearfish, understanding such a rapid temperature change didn't just intrigue historians; it provided crucial insights into meteorological extremes. People often utilize temperature calculations to assess whether conditions are suitable for outdoor events, providing a comfortable experience based on the expected temperature outcomes.

Real-world usage of these calculations includes:

• Adjusting heating or cooling systems based on forecasted temperatures.
• Planning travel by predicting weather conditions during a trip.
• Helping farmers decide the best time for planting and harvesting crops.

In essence, the ability to accurately compute such temperature shifts isn't just an academic exercise, but a tool for everyday decision-making and planning.