Problem 127 - Algebra A Combined Function

Question

Translate each phrase to an expression. Use \(x\) to represent "a number." A deep-sea diver must move up or down in the water in short steps in order to keep from getting a physical condition called the "bends." Suppose a diver moves down from the surface in five steps of 20 feet each. Represent his total movement as a product of signed numbers and find the product.

Step-by-Step Solution

Answer

The diver moves a total of -100 feet.

Step 1: Recognize the Variable

First, identify what 'a number' or the unknown quantity in the problem is referring to. Here, the diver's movement per step is "20 feet," which is represented as a number. Thus, 'a number' is consistently 20 in each step.

Step 2: Understand the Direction

The diver is moving downwards, which in terms of signed numbers, is represented by negative values. Therefore, each step of 20 feet down is represented by -20.

Step 3: Calculate Number of Steps

The problem states the diver moves in five steps of 20 feet each. Therefore, these five steps need to be considered in our expression for multiplying signed numbers.

Step 4: Formulate the Expression

To find the total movement, multiply the distance of each step (-20) by the number of steps (5): \[\text{Total movement} = x \cdot = (-20) \cdot 5\]

Step 5: Compute the Product

Calculate the product of the numbers obtained in the previous step. Here, multiply -20 by 5: \[\text{Product} = (-20) \times 5 = -100\]

Step 6: Interpret the Result

The product, -100, indicates that the diver moved a total of 100 feet downward from the surface.

Key Concepts

Signed Numbers

Signed numbers refer to numbers that have a positive or negative sign before them. Understanding signed numbers is crucial for expressing changes in direction or quantity, especially when dealing with problems involving direction like the diver's movement. In mathematics, a positive number is usually written without a sign or with a "+" sign (e.g., 20 or +20), while a negative number is always accompanied by a "-" sign (e.g., -20).

• **Positive numbers:** Indicate a value above zero or movement in a forward direction.
• **Negative numbers:** Indicate a value below zero or movement in a backward or downward direction.

In the exercise, because the diver is moving downwards, each step is represented as a negative number (-20), thus capturing both the concept of direction and magnitude.

Variables in Expressions

Variables are symbols used to represent unknown values or quantities in mathematical expressions. They help us create general formulas that solve a wide range of problems. Typically, variables are denoted by letters such as \( x \), \( y \), or \( z \), and in this case, "a number" is to be represented by a variable.Consider the expression from our problem: \[ \text{Total movement} = (-20) \cdot 5 \]Here, the variable could represent the diver's step distance in a more flexible equation. However, since the problem specifies the number as 20 feet, we directly substituted \( x = 20 \) in each calculation. Even in such straightforward cases, recognizing the role of a variable strengthens problem formulation and solving skills. It aids in breaking down complex problems as simply as recognizing 'a number'.Variables are powerful tools, allowing students to plug different values into an equation and see how outcomes change.

Multiplication of Integers

Multiplying integers, which can be positive or negative whole numbers, follows straightforward rules that can be easy to remember.

• **Positive x Positive = Positive**
• **Negative x Negative = Positive**
• **Positive x Negative = Negative**
• **Negative x Positive = Negative**

In the exercise, we multiply -20 (negative) by 5 (positive). According to the rules above, a positive number times a negative number yields a negative result. Hence, \[ (-20) \times 5 = -100 \]This result, \(-100\), meaning the diver moved 100 feet in the downward direction from the surface. Understanding these multiplication rules helps simplify calculations and aids in correctly interpreting the results of such exercises.