Problem 40 - Algebra 1

Question

State whether the variables have direct variation, inverse variation, or neither. The mass \(m\) and the volume \(V\) of a substance are related by the equation \(2 V=m,\) where 2 is the density of the substance.

Step-by-Step Solution

Answer

The variables mass \(m\) and volume \(V\) of a substance have a direct variation.

Step 1 - Identify the Equation

First of all, identify the equation given. Here, the equation is \(2V = m\), relating the mass \(m\) and volume \(V\) of a substance.

Step 2 - Isolate the Variables

Rewrite the equation to isolate the variables. It can be rewritten as \(m = 2V\). Here 2 is a constant, representing the density of the substance.

Step 3 - Analyze Variation Type

Based on the form \(y = kx\), where \(y\) is dependent variable, \(x\) is independent variable, and \(k\) is constant of variation, it can be inferred that it is a direct variation as the equation is in compatible form and both variables are positively related. As the volume \(V\) increases, the mass \(m\) also increases, and vice versa.

Key Concepts

Understanding Mass and Volume Relationship

The relationship between mass and volume is fundamental in understanding the properties of substances. Mass is the amount of matter in an object, while volume measures the space it occupies. Together, they help to determine how dense a substance is.
In the given exercise, the equation \(m = 2V\) expresses the connection between mass \(m\) and volume \(V\). Here, the mass of a substance is directly proportional to its volume.
This means that if you double the volume of the substance, its mass will also double. Such a relationship is often referred to as direct variation because both variables change in the same direction.

Exploring Density

Density is a key property that relates mass and volume. It tells us how tightly matter is packed in a substance. Mathematically, density \(d\) is defined as the mass \(m\) of a substance divided by its volume \(V\). This can be written as:

• Density \(d = \frac{m}{V}\)

In our exercise, the constant "2" in the equation \(m = 2V\) represents the density. It indicates that for every unit of volume, there are 2 units of mass. If density is high, the substance is compact; if it's low, the substance is less dense.
This concept is essential in various real-world applications, such as determining whether objects will float or sink in water.

Using Mathematical Equations to Determine Variations

Mathematical equations are powerful tools to determine the type of variation between variables. In the exercise, the equation \(m = 2V\) follows the form \(y = kx\). Here, \(k\) is the constant of proportionality, or in this case, the density.
Direct variation occurs when one variable increases, the other also increases at a constant rate, which is clearly shown in the equation. Such equations often appear in mathematics and science to describe real-world phenomena. Their simplicity makes it easy to predict how changing one variable will affect the other.
By mastering these equations, students can analyze and solve various problems involving direct relationships efficiently. This lays a strong foundation for more advanced topics in physics and chemistry.