Problem 39 - Algebra 1

Question

State whether the variables have direct variation, inverse variation, or neither. The area \(B\) of the base and the height \(h\) of a prism with a volume of 10 cubic units are related by the equation \(B h=10\).

Step-by-Step Solution

Answer

The variables \(B\) and \(h\) exhibit inverse variation.

Step 1: Identify the Type of Equation

The equation \(Bh=10\) is given. If we isolate one variable, it will become clearer what type of variation we are dealing with. Let's rearrange this equation by dividing both sides by \(h\), such that \(B = 10/h\).

Step 2: Match with Variation Type

The isolated equation \(B = 10/h\) is in the form of \(B = k/h\), which matches the formula of inverse variation \(xy = k\), with \(B\) representing \(x\), \(h\) representing \(y\), and \(10\) representing \(k\).

Step 3: Determine Variation Type

As the equation \(B=10/h\) matches the form of inverse variation, it follows that the variables \(B\) and \(h\) exhibit inverse variation.

Key Concepts

Direct Variation

An essential concept in algebra is direct variation, which describes a relationship between two variables in which one is a constant multiple of the other. In mathematical terms, when two variables, let's call them \( x \) and \( y \), show direct variation, we can express their relationship with the equation \( y = kx \), where \( k \) is a non-zero constant known as the coefficient of variation.

For example, if the number of items you buy and the total cost directly vary, buying more items will increase the cost directly in proportion to the quantity. This kind of relationship is predictable and linear, which means that if you were to create a graph with \( x \) and \( y \), you would see a straight line passing through the origin (0,0). Understanding direct variation is crucial when trying to recognize patterns of change in real-world contexts, such as speed and travel time, or the classic distance-rate-time relationships.

Prism Volume

The volume of a prism is a fundamental topic in geometry, particularly in understanding three-dimensional shapes. A prism is a solid object with two congruent and parallel faces, and its volume is calculated by finding the product of the area of the base \( B \) and the height \( h \) of the prism. Mathematically, we express this as \( V = Bh \), where \( V \) represents the volume.

To find a prism's volume, you first need to calculate the area of the base, which can be shaped like any polygon, and then multiply it by the prism's height. The unit of measurement is typically cubic units since volume measures the three-dimensional space an object occupies. Understanding the principle behind prism volume is vital in various applications, from packaging and construction to manufacturing and design.

Variation Types in Algebra

In algebra, understanding different types of variation relationships between variables allows us to model and solve real-world problems. Apart from direct variation and its counterpart, inverse variation, there are other significant types to consider:

• Joint Variation: A situation where a variable is directly proportional to the product of two or more other variables.
• Combined Variation: A scenario where a variable varies directly with one set of variables and inversely with another.
• Partial Variation: A form of variation that includes both direct variation and a fixed value, which is represented in the equation \( y = mx + b \), where \( b \) is not zero.

Each type of variation has unique characteristics that serve different purposes when modeling relationships between quantities. For instance, inverse variation is often used in situations where one quantity increases as another quantity decreases, such as the intensity of light varying inversely with the square of the distance from the light source. Recognizing these patterns is key to problem-solving in not just mathematics but also in fields that rely on quantitative analysis, like economics, engineering, and the physical sciences.