Problem 9 - Precalculus Mathematics for Calculus

Question

Write an equation that expresses the statement. \(y\) is proportional to \(s\) and inversely proportional to \(t\)

Step-by-Step Solution

Answer

The equation is \( y = k \cdot \frac{s}{t} \).

Step 1: Understand Proportional Relationships

In mathematics, when a quantity is proportional to another, it means they change together at the same rate. Here, the statement says that \( y \) is proportional to \( s \) meaning \( y = k \cdot s \) where \( k \) is a constant of proportionality.

Step 2: Understand Inverse Proportionality

For a quantity to be inversely proportional to another means as one increases, the other decreases. In this case, \( y \) is inversely proportional to \( t \), which can be expressed as \( y = \frac{k}{t} \).

Step 3: Combine Proportional and Inverse Proportional Relationships

Since \( y \) is both directly proportional to \( s \) and inversely proportional to \( t \), we combine these relationships into a single equation: \( y = k \cdot \frac{s}{t} \).

Step 4: Expression of the Equation

The complete equation expressing the given statement is \( y = k \cdot \frac{s}{t} \) where \( k \) is a constant that links the proportional and inverse relationships.

Key Concepts

Direct Variation

In direct variation, two quantities increase or decrease together at a constant rate. When one quantity changes, the other does so in direct proportion. If you double one quantity, the other one also doubles. For this relationship, the formula used is \( y = k \cdot x \), where:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( k \) is the positive constant of proportionality.

The constant \( k \) is crucial because it represents the rate at which \( y \) changes with respect to \( x \). For example, if you have a situation where the distance \( y \) traveled is proportional to the time \( x \), the constant \( k \) would represent speed, showing how fast the distance changes as time passes.
Direct variation is straightforward, and it forms a linear relationship that passes through the origin \((0, 0)\) when plotted on a graph.

Inverse Variation

Inverse variation describes a scenario where as one quantity increases, the other decreases at an inversely proportional rate. Hence, the two variables are not changing at the same rate. In mathematical terms, if \( y \) varies inversely with \( x \), the formula is given by \( y = \frac{k}{x} \). Here:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( k \) is a positive constant.

The idea is similar to a seesaw, where as one side goes up, the other comes down. For instance, if you are driving a fixed distance \( y \), the time \( x \) needed decreases as your speed \( k \) increases.
This relationship does not pass through the origin like direct variation, but it forms a hyperbolic curve on a graph, trending towards the axes without touching them.

Constant of Proportionality

The constant of proportionality is a key component in both direct and inverse variations. It is the constant \( k \) that defines the specific rate of change between the two variables. Without this constant, you cannot accurately describe how one variable affects another.
This constant acts as a bridge between the variables:

• In direct variation, \( k \) tells us how fast or slow \( y \) changes as \( x \) changes: \( y = k \cdot x \).
• In inverse variation, \( k \) indicates the product \( y \, x \) remains constant: \( y = \frac{k}{x} \).

Understanding \( k \) means understanding the efficiency or ratio inherent in a system:

• For direct variation, it could represent things like speed (change in distance over time) or density (mass per volume).
• In inverse variation, it could represent the time needed to complete a task at varying speeds.

Recognizing and calculating this constant is crucial for predicting outcomes and modeling situations using mathematical expressions.