Problem 3 - Intermediate Algebra

Question

Translate each statement of variation into an equation, and use \(k\) as the constant of variation. \(C\) varies directly as \(g\) and inversely as the cube of \(t\).

Step-by-Step Solution

Answer

Equation: \(C = k \frac{g}{t^3}\).

Step 1: Understanding Direct Variation

When a variable varies directly as another variable, it means that as one variable increases, the other increases at a constant rate. If variable \(C\) varies directly as \(g\), the relationship can be expressed as \(C = kg\), where \(k\) is the constant of variation.

Step 2: Understanding Inverse Variation

Inverse variation implies that one variable increases while the other decreases. If \(C\) varies inversely as the cube of \(t\), the relationship can be given as \(C = \frac{k}{t^3}\). Again, \(k\) denotes the constant of variation here.

Step 3: Combining Both Variations

In this problem, \(C\) varies directly as \(g\) and inversely as \(t^3\). To combine these two variations, we write the equation as \(C = k \frac{g}{t^3}\). This equation reflects both the direct variation with \(g\) and the inverse variation with the cube of \(t\).

Key Concepts

Direct Variation

Direct variation refers to a relationship between two variables where they increase or decrease together at a constant rate. Imagine two friends walking at the same pace; as one covers a certain distance, the other covers the same distance. In mathematical terms, direct variation can be expressed with the equation \(y = kx\). Here, \(y\) varies directly with \(x\), and \(k\) is the constant of variation, determining how much they change together. Understanding direct variation is crucial for solving problems where changes in one variable affect another proportionally.

Inverse Variation

Inverse variation describes a relationship where the increase of one variable results in the decrease of another, and vice versa. Think of a teeter-totter: as one side goes up, the other goes down. Mathematically, inverse variation can be represented by the equation \(y = \frac{k}{x}\). In this scenario, \(y\) varies inversely with \(x\), and \(k\) is again the constant of variation. This relationship is key in systems where two quantities behave oppositely.

Constant of Variation

The constant of variation, denoted as \(k\), simplifies complex relationships by providing a fixed multiplier or divisor between variables. It's like a bridge that connects the two sides of an equation. In direct variation, \(k\) indicates how closely related two increasing variables are, as in \(y = kx\). For inverse variation, \(k\) measures the extent to which one variable inversely affects another, as seen in \(y = \frac{k}{x}\). Having a firm grasp of these constants aids in predicting and understanding variable behavior.

Algebraic Equations

Algebraic equations are mathematical statements that show the equal balance between two expressions. They often include variables, numbers, and operations, such as addition or multiplication. For example, understanding the equation \(C = k \frac{g}{t^3}\) becomes crucial when solving for one variable based on the values of others. By manipulating algebraic equations, students can solve for unknowns and better understand complex relationships in mathematics. The ability to translate word problems into these equations forms the basis of algebraic problem solving.