Problem 59 - Algebra 1: Concepts and Skills

Question

What is the velocity of a diver who descends to the ocean floor at a rate of 3 meters per second? \begin{equation}|-3| \mathrm{m} / \mathrm{sec} \end{equation}

Step-by-Step Solution

Answer

The velocity of the diver is \(3 \: m/sec\).

Step 1: Identify the given velocity

The velocity given in the exercise is -3 m/sec. The negative sign indicates that the diver is moving downwards.

Step 2: Compute the absolute value

To calculate the absolute value of -3 m/sec, remove the negative sign. Thus, the velocity becomes 3 m/sec.

Key Concepts

Velocity in Mathematics

When discussing velocity in mathematics, we're diving into one of the foundational concepts in physics and mathematics, especially when objects are in motion. Velocity refers to the speed of an object in a given direction. It is a vector quantity which means that it has both magnitude and speed.

In the exercise provided, the velocity of a diver who descends to the ocean floor at a rate of 3 meters per second is examined. To understand the diver's velocity, we must consider the direction of the movement along with its speed. The negative value (-3 m/sec) signifies that the diver is moving in a direction opposite to what we might consider the positive direction (typically upwards or forwards).

However, velocity's magnitude is treated as a positive quantity when we merely wish to express the speed without concern for direction. Therefore, saying the diver's velocity is 3 meters per second is correct when we're focusing on how fast the diver is moving, regardless of the direction being downwards.

Absolute Value in Algebra

The concept of absolute value in algebra is pivotal when dealing with real numbers, as it represents the distance of a number from zero on the number line, ignoring the direction. The absolute value of a number is always non-negative.

In algebra, the absolute value is indicated by two vertical bars surrounding the number, like this: \( |x| \). For our diver's velocity, the absolute value is computed as \( |-3| \), which equals 3. This is because the absolute value of -3 is the distance between -3 and 0 on the number line, which is simply 3 units, irrespective of the negative sign.

Understanding absolute values is essential, because it’s used in various contexts such as determining the difference between numbers, formulating equations that account for magnitude only, and simplifying complex mathematical expressions where direction is not pertinent.

Negative Numbers in Algebra

When it comes to negative numbers in algebra, we often think about what lies below zero on the number line. They are used to represent opposites or a deficiency of something. In the context of velocity, a negative number usually indicates a direction that is opposite to the positive direction, as mentioned earlier.

For the diver descending to the ocean floor, the velocity given is -3 m/sec. This negative sign conveys that the diver is moving downward, which is conventionally taken as the negative direction. The role of negative numbers in algebra is significant because it helps in portraying real-world situations that involve loss, decrease, or reverse movements.

It’s important for students to recognize that the idea of direction is central in the application of negative numbers. The concept broadens the ability to express a wide range of mathematical ideas, from simple calculations to complex functions and equations.