Problem 88 - Algebra A Combined Function

Question

Solve. See Example 22. 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 \(45^{\circ}\text{F}\) by 7:32 a.m.

Step 1: Initial Temperature

Identify the starting temperature. Initially, at 7:30 a.m., the temperature is \( -4^{\circ}\text{F} \).

Step 2: Temperature Increase

Determine the temperature increase. The temperature increased by \( 49^{\circ}\text{F} \) in the next 2 minutes.

Step 3: Calculate New Temperature

Add the temperature increase to the initial temperature to find the new temperature at 7:32 a.m. \(-4^{\circ}\text{F} + 49^{\circ}\text{F} = 45^{\circ}\text{F}\).

Key Concepts

Understanding Initial Temperature

When we talk about the initial temperature, we are referring to the starting point before any changes occur. It's the baseline temperature measurement from which changes are calculated. In our example, the initial temperature in Spearfish, South Dakota, was \(-4^{\circ}\text{F}\) at 7:30 a.m. This negative value indicates that it was below freezing point, highlighting how cold it was at that time.
Understanding the initial temperature is crucial because it sets the stage for determining how much the temperature has changed. By knowing this baseline, we can accurately measure subsequent increases or decreases in temperature.
The initial temperature serves as a reference to assess change. Whether it's a cold winter morning or a hot summer afternoon, knowing the initial temperature helps us understand the environmental conditions at a specific moment.

What is Temperature Increase?

A temperature increase refers to a rise in temperature over a period of time. It helps us quantitatively measure how much warmer a location has become. In our scenario, the temperature in Spearfish increased by \(49^{\circ}\text{F}\) in just 2 minutes, which is quite remarkable!
Knowing the amount of temperature increase is key to understanding the sudden shifts in weather conditions, such as unexpected warm fronts. This measurement is the difference between the initial and later temperatures.
Temperature increase can indicate changing weather patterns. Monitoring these increases helps meteorologists predict weather conditions and understand climate trends.

• Track temperature changes for planning daily activities.
• Helps gauge if conditions are becoming more favrable or dangerous.

Recognizing significant temperature increases can be crucial for making informed decisions about safety and comfort.

Arithmetic Addition for Temperature Changes

Arithmetic addition is a simple mathematical process we use to combine numbers. In the context of temperature changes, addition helps us determine the new temperature after a change has occurred.
In our example, by adding the initial temperature ( \(-4^{\circ}\text{F}\) ) to the temperature increase ( \(49^{\circ}\text{F}\) ), we calculate the new temperature. Let's see how it's done: \(-4^{\circ}\text{F} + 49^{\circ}\text{F} = 45^{\circ}\text{F}\).
This simple operation shows how far above or below the initial value the temperature has moved. By applying arithmetic addition:

• We efficiently find out the current conditions.
• Allows prediction of future states by adding expected changes.

Arithmetic addition is vital for quick problem-solving, whether you're dealing with temperatures or any other numerical information. It is an essential skill that provides clarity on how quantities change over time.