Problem 85 - Algebra A Combined Function

Question

Solve. See Example 22. Suppose a deep-sea diver dives from the surface to 215 feet below the surface. He then dives down 16 more feet. Use positive and negative numbers to represent this situation. Then find the diver's present depth.

Step-by-Step Solution

Answer

The diver's present depth is 231 feet below the surface.

Step 1: Understand the Problem Context

The problem involves a deep-sea diver diving below the surface of the ocean. Each depth can be represented using negative numbers, since the diver is going below the surface.

Step 2: Represent the Initial Dive

The diver initially dives 215 feet below the surface. We represent this as a depth of \(-215\) feet, where negative indicates below the surface.

Step 3: Represent the Additional Dive

The diver then dives an additional 16 feet deeper. This is also below the surface, so it is represented as \(-16\) feet.

Step 4: Calculate the Present Depth

To find the diver's current depth, we add the two depths together: \(-215 + (-16)\). Adding two negative numbers involves adding their absolute values and keeping the negative sign.

Step 5: Perform the Addition

Calculate \(-215 + (-16)\): \[ -215 + (-16) = -215 - 16 = -231 \] The result is \(-231\), indicating the diver is 231 feet below the surface.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, used to represent a decrease or opposite direction compared to positive numbers. In mathematics, a negative number is usually found to the left of zero on a number line. Negative numbers let us denote things like temperatures below freezing or depths below sea level.
When you see a minus sign in front of a number, it simply means it’s a negative number, such as \(-3\) or \(-215\). These numbers play an important role in various applications, such as measuring depth or financial loss.

• Used in real-life contexts: temperature, altitudes, and financial operations.
• On a number line, they are to the left of zero.
• In problems involving below-surface measurement, negative numbers are typically used.

Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. This means that whether a number is positive or negative, its absolute value is always positive. For example, the absolute value of both \(3\) and \(-3\) is \(3\).
In mathematical terms, the absolute value of a number \(x\) is written as \(|x|\). It answers the question "How far is the number from zero?"

• Always positive, or zero.
• Expressed as \(|x|\).
• Helps in operations involving negative numbers, such as addition.

Addition of Integers

Adding integers involves determining whether the integers are positive or negative. When both integers are negative, you add their absolute values and place a negative sign in front of the result.
This is exactly what happens when we solve the problem of a diver's depth, represented by \(-215 + (-16)\). You can think of this as:

• Adding \(215\) and \(16\), which equals \(231\).
• Since both original numbers were negative, the result is negative.

Thus, \(-215 + (-16) = -231\). This method allows us to combine negative numbers effectively, ensuring we maintain the context of the situation.

Depth Measurement in Mathematics

Depth measurement in mathematics often uses negative numbers to signify a position below a fixed reference point, such as sea level. This convention helps to easily distinguish between locations above and below this reference point.
In our problem, diving below the sea surface is represented with negative numbers. Understanding this concept helps when solving problems that involve vertical positions, such as altitude or elevation, ensuring accuracy in representation and calculations.

• Negative values are common in contexts like diving and tunneling.
• Typically measured from a reference point like sea level.
• Used alongside other mathematical operations to determine net changes in depth.