Problem 92 - Algebra for College Students

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 and then continued to dive down for another 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: Determine Initial Position

The scuba diver starts at 32 feet below sea level. This can be represented as -32 feet relative to sea level.

Step 2: Calculate Ascent

The diver ascends 13 feet to meet his partner. Ascending means moving closer to sea level, so we add 13 feet to the initial position: \[-32 + 13 = -19 \]This means the diver is at -19 feet below sea level after meeting his partner.

Step 3: Calculate Additional Dive

After meeting his partner, the diver continues to dive down 50 feet further. Diving down means moving away from sea level, so we subtract 50 feet from his position:\[-19 - 50 = -69 \]Thus, the diver is 69 feet below sea level at the end of his dive.

Key Concepts

Integer Operations

Integers are whole numbers that can be positive, negative, or zero. When performing operations including addition and subtraction with integers, it is crucial to consider their signs. In the context of this scuba diving problem, integer operations help track the diver's movements relative to sea level.

In the problem, you start with the diver's initial position. This is indicated by the integer -32 feet, representing 32 feet below sea level. Moving upwards involves adding a positive value, which brings the diver closer to the zero level, meaning the surface of the sea.

When the diver ascends 13 feet, the new position is calculated by the equation: \[ -32 + 13 = -19 \]
The result (-19) signifies the diver is now 19 feet below sea level. These simple mathematics manage our understanding of real-world directions and depths using integer operations.

• Integers can represent both directions, such as up/down.
• Addition with integers can move positions closer (up if starting negative) or further apart.
• Subtraction leads to further decreasing a position (more depth).

Real-World Applications

Mathematics plays a vital role in various real-world situations, including those involving physical movements and navigation like scuba diving. Understanding integer operations allows divers, and others working in similar fields, to calculate their positions efficiently.

Knowing how to read and calculate depth can prevent dangerous situations within underwater exploration environments. The diver’s calculations help monitor or plan for conditions, ensuring their safety and maintaining awareness of their physical limits. This shows the importance of understanding and performing accurate calculations under different sea conditions.

• Diving requires precise calculations of depth and movement.
• Integer calculations can inform situational awareness and safety protocols.
• Real-world scenarios demonstrate the necessity of mathematical accuracy.

Word Problems

Word problems in mathematics are statements that require translation into mathematical expressions to find a solution. They often relate to real-life scenarios, making their problems and solutions more understandable and relatable.

Breaking down the elements of a word problem involves identifying each component, such as initial conditions and changes that occur through actions. For example, in the scuba diver’s case:

• Initial Condition: Start at -32 feet below sea level.
• Action 1: Ascend 13 feet to meet partner.
• Action 2: Dive an additional 50 feet.
• Finally: Calculate the diver’s new position by performing necessary integer operations.

This breakdown helps convert the described scenario into mathematical expressions, facilitating a solution. Understanding how to interpret and solve word problems is crucial for mathematics since it bridges mathematical concepts with practical use cases.