Problem 86 - Algebra A Combined Function

Question

Solve. See Example 22. Suppose a diver dives from the surface to 248 meters below the surface and then swims up 8 meters, down 16 meters, down another 28 meters, and then up 32 meters. Use positive and negative numbers to represent this situation. Then find the diver's depth after these movements.

Step-by-Step Solution

Answer

The diver's depth is 252 meters below the surface.

Step 1: Initial Position

The diver starts at the surface of the water. We represent this as 0 meters.

Step 2: Dive to Initial Depth

The diver dives 248 meters below the surface. We represent this as \[-248 \text{ meters}\].

Step 3: First Ascent Calculation

The diver swims up 8 meters from the initial depth. To find the new depth, we calculate: \[-248 + 8 = -240 \text{ meters}\].

Step 4: First Descent Calculation

Next, the diver swims down 16 meters. We further adjust the depth: \[-240 - 16 = -256 \text{ meters}\].

Step 5: Second Descent Calculation

The diver then swims down another 28 meters. Update the depth again: \[-256 - 28 = -284 \text{ meters}\].

Step 6: Second Ascent Calculation

Finally, the diver swims up 32 meters. To find the final depth, we calculate: \[-284 + 32 = -252 \text{ meters}\].

Key Concepts

Positive and Negative Numbers

In mathematics, positive and negative numbers are used to describe values with direction, which often involves comparing values above and below a certain reference point. In our everyday life, these numbers help explain situations like temperature changes, financial profits and losses, and movements above or below sea level.
When discussing deep-sea diving, we use positive numbers to represent movements upwards or increases, and negative numbers to represent dives or decreases below the surface. In the exercise, each movement of the diver is expressed using these numbers. Thus when the diver swims up 8 meters, it’s represented as "+8". Conversely, when the diver goes down 16 meters, it becomes "-16".
This conceptualization is essential because operations with positive and negative numbers follow specific rules. For instance, subtracting a negative number equates to adding its positive counterpart. This is akin to changing directions in real life and understanding this is fundamental to solving many integer arithmetic problems.

Depth Calculation

Depth calculation is a practical application of integer arithmetic, where positive and negative numbers are used to track changes in depth.
Let's break down the diver's journey:

• Initial depth is 0 meters at the surface.
• The diver descends 248 meters, resulting in a depth of -248 meters.
• He swims up by 8 meters to reach -240 meters.
• A further descent of 16 meters brings him to -256 meters.
• Adding another 28 meters downwards, his new depth becomes -284 meters.
• Finally, swimming 32 meters upwards makes his final depth -252 meters.

Understanding depth calculation through integer arithmetic allows easy monitoring of how far below or above a reference point one is, using consecutive additions and subtractions. This structuring provides clarity when managing real-world tasks involving levels of depth.

Real-World Application of Integers

The use of integers extends far beyond theoretical mathematics and into real-world scenarios. They provide a straightforward yet powerful means of representing situations involving gains and losses, heights and depths, or any circumstances where comparison against a baseline occurs.
In diving scenarios such as the example exercise, integers effectively describe movements both above and below the sea surface, making tracking accurate and calculations simple.
Beyond diving, integers play crucial roles in other fields:

• In finance, they represent profits as positive numbers and debts as negative numbers.
• In meteorology, temperature changes above and below freezing are denoted similarly.
• Geographically, altitudes above or below sea level are measured in this manner.

By understanding how integers can be applied in various contexts, students and professionals can gain clearer insights into everyday phenomena, optimizing decision-making and problem-solving abilities.