Problem 31 - Algebra 1: Concepts and Skills

Question

In Exercises \(31-33,\) state whether the variables model direct variation, inverse variation, or neither. BASE AND HEIGHT 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 given equation \(Bh = 10\) models an inverse variation.

Step 1: Identifying the variation

Looking at the equation \(Bh = 10\), notice that when you multiply B (base area) and h (height), it always equals 10, which is a constant. In this case, both variables are on the same side of the equation, and they are multiplied together.

Step 2: Applying the definition of direct and inverse variation

In a direct variation, when one variable increases the other also increases, while in an inverse variation when one variable increases the other decreases. However, in both cases the product of the two variables is constant. So the key factor here is whether both variables increase or decrease together (direct variation), or if one increases while the other decreases (inverse variation).

Step 3: Classifying the variation

In the given equation \(Bh = 10\), if the base area B increases then the height h must decrease for the equation to hold true, and vice versa. Thus, the variables are inversely related. As such, the equation \(Bh = 10\) is an example of inverse variation.

Key Concepts

Direct Variation Explained

Direct variation is a relationship in which one variable increases or decreases proportionally with another variable. When variables are in direct variation, they can be represented by the equation \( y = kx \), where \( y \) and \( x \) are the variables, and \( k \) is the constant of variation. This means that as \( x \) increases, \( y \) increases by a constant factor, and vice versa.
This concept is often seen in real-world situations where scaling factors play a crucial role, such as the relation between time and speed to cover a fixed distance.

• If \( x \) doubles, \( y \) doubles as well.
• The ratio \( \frac{y}{x} \) remains constant, which is the constant \( k \).
• Imagine if the price per kilogram of apples is fixed; the total cost (\( y \)) varies directly with the weight in kilograms (\( x \)).

Understanding direct variation can help in recognizing how variables change together in a predictable way, which is useful in both algebraic problems and practical situations.

Prism Volume Demystified

The concept of prism volume is all about understanding how much space a prism occupies. A prism is a solid object with two identical ends and flatsides. The volume of a prism is calculated by multiplying the area of its base (\( B \)) by its height (\( h \)). The formula is:
\[ Volume = B \times h \]
Prisms can have various forms, such as rectangular, triangular, or other shapes as their bases. But regardless of the shape of the base, the formula for volume remains the same:

• The base area \( B \) is determined by the shape of the base; for example, in a rectangular prism, it's \( length \times width \).
• The height \( h \) is the perpendicular distance between the two parallel bases.
• Imagine filling a box (rectangular prism) with water; its volume would be how much water it can hold.

Knowing how to calculate prism volume is helpful in fields like architecture, packaging, and any scenario where determining capacity is important.

Algebraic Equations Simplified

Algebraic equations form the backbone of algebra and are used to express mathematical relationships. An equation consists of variables, constants, and arithmetic operations. It states that two expressions are equal, often requiring finding the value of unknown variables. A simple form of an algebraic equation is:
\[ ax + b = c \] where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.
Algebraic equations can be linear, quadratic, or of higher order, but linear equations are particularly common in everyday problem-solving. Solving these equations involves:

• Isolating the variable to one side of the equation.
• Performing inverse operations to simplify the equation.
• Checking solutions by substituting back into the original equation.

Mastering algebraic equations enables you to tackle problems in physics, engineering, economics, and beyond by representing and solving real-world situations mathematically.