Problem 32 - Algebra 1: Concepts and Skills

Question

In Exercises \(31-33,\) state whether the variables model direct variation, inverse variation, or neither. MASS AND VOLUME 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 model a direct variation.

Step 1: Understand Direct and Inverse Variation

In a direct variation, two quantities increase or decrease together in such a way that their ratio is constant. This can be represented by the formula \(y = kx\), where \(k\) is the constant of variation. In an inverse variation, one quantity decreases as the other increases, or vice versa, such that their product is constant. This can be represented by the formula \(xy = k\), where \(k\) is the constant of variation.

Step 2: Analyze the Given Equation

The equation given is \(2V = m\). This can be rearranged to \(m = 2V\). This equation shows that as \(V\) (volume) increases or decreases, \(m\) (mass) also increases or decreases by the same factor, indicating that volume and mass directly vary.

Step 3: Determine the Type of Variation

Since the equation is \(m = 2V\), and matches the form of direct variation \(y = kx\), with density \(2\) acting as the constant of variation, it can be concluded that the variables model a direct variation.

Key Concepts

Mass

Mass is a fundamental property of matter. It measures the amount of substance contained within an object. Think of mass as the weightiness or the heaviness of an object, but without considering gravity. This means, while weight can change depending on where you are (like on the moon or Earth), mass stays constant because it's an intrinsic property. To further simplify:

• Mass is usually measured in kilograms (kg) or grams (g).
• It does not change regardless of location, unlike weight.
• Mass is a measure of the quantity of matter.

So, when we talk about mass increasing or decreasing, we are talking about adding or removing the substance from the object. When mass is in direct variation with volume, like in the formula given, it means as volume increases, mass does as well—assuming density stays the same.

Volume

Volume refers to the space an object or substance occupies. It's like a container's capacity. Imagine pouring water into a bottle; as you pour more, the liquid fills more space within the container—that's volume. Here’s some useful information about volume:

• Volume is usually measured in liters (L), cubic meters (m³), or cubic centimeters (cm³).
• A larger volume means more space is occupied.
• It can change with changes in pressure or temperature, especially for gaseous substances.

In the context of direct variation with mass, volume is directly proportional to mass when density remains constant. That means increasing the volume will increase the mass proportionally, as shown in the equation 2V = m. This highlights a direct relationship where the factor of density remains unchanged.

Density

Density is the measure of how much mass is contained within a certain volume. It's the ratio of mass to volume, typically expressed as mass per unit volume. This is important because it helps us understand how tightly packed the substance’s molecules are.Important points about density include:

• It is often given in units of kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³).
• Density is a key factor in determining whether objects float or sink in a liquid.
• It's calculated using the formula: \[ \text{Density} = \frac{m}{V} \]

In the exercise’s equation 2V = m, the constant 2 represents the density. This constant reveals that for every unit of volume, there is twice the mass, highlighting the direct relation between mass and volume through density.