Problem 96 - Intermediate Algebra

Question

Write a numerical statement to represent the problem. Then simplify the numerical expression to answer the question. A scuba diver was 32 feet below sea level when he noticed that his partner had his extra knife. He ascended 13 feet to meet his partner, get the knife, and then dove down 50 feet. How far below sea level is the diver?

Step-by-Step Solution

Answer

The diver is 69 feet below sea level.

Step 1: Understand the Problem

The problem describes the movements of a scuba diver in relation to sea level, which is our reference point. We'll track the diver's positions using positive and negative numbers: negative numbers for movements below sea level and positive numbers for movements above sea level.

Step 2: Write the Initial Numerical Statement

We need to write the diver's movements as a numerical expression. Initially, the diver was at \(-32\) feet (below sea level), ascended 13 feet (\(+13\)), and then descended 50 feet (\(-50\)). The numerical statement representing this sequence is:\[ -32 + 13 - 50 \]

Step 3: Simplify the Expression Part 1

Start by simplifying the first part of the expression. Calculate the new position after ascending 13 feet:\[ -32 + 13 = -19 \]So the diver is at \(-19\) feet after the ascent.

Step 4: Simplify the Expression Part 2

Now, calculate the diver's position after descending 50 feet further:\[ -19 - 50 = -69 \]The diver is now at \(-69\) feet.

Key Concepts

Addition and Subtraction

When solving problems involving numerical expressions, it's crucial to follow the proper steps for addition and subtraction. These operations form the foundation for determining changes in positions, like those in our marine examples. In the scuba diver scenario, we see a combination of adding and subtracting numbers to determine position changes.

Here's a quick guide to handle such scenarios:

• Always start with your initial position. Here, it's the diver at -32 feet below sea level.
• Add the change in position when the diver moves up, as positive numbers represent ascents. In this case, adding 13 steps brings the diver to -19 feet.
• Subtract when the diver moves down further below the initial position. Subtracting the 50-foot dive concludes at -69 feet below sea level.

Tackling these steps methodically ensures accuracy in problem-solving.

Negative Numbers

Working with negative numbers can seem daunting at first, but they are quite powerful when understanding movement relative to a fixed point, such as sea level. Negative numbers denote positions below the reference point, which is quite useful in vertical movement cases like diving or descending.

Consider the following essential points:

• Negative numbers are values less than zero and indicate depth or decrease in real-world scenarios.
• When adding a positive number to a negative, it moves closer to zero, potentially becoming positive if the addition is large enough.
• Adding two negative numbers increases or furthers the negative state, or in other words, it acts more like subtraction.

In our example, all diver's movements below sea level are tagged as negative, providing a clear map of dives and ascents.

Sea Level

Sea level serves as a crucial reference point, particularly in scenarios involving water bodies and their depths. It represents a boundary between above water (positive numbers) and underwater (negative numbers).

Why sea level matters:

• It standardizes depth measurement, making it easier to quantify and communicate changes.
• Positions above sea level use positive numbers indicating height above, while below uses negative values.
• Understanding sea level helps visualize these vertical movements logically and numerically.

In our scuba diving problem, sea level is where the diver begins and makes sense of the ascent and descent operations described in the numerical expressions. Understanding the diver's expedition is easier when we visualize his transition relative to this baseline.