Problem 49 - Precalculus with Limits

Question

In Exercises 49-58, find a mathematical model for the verbal statement. \(A\) varies directly as the square of \(r\).

Step-by-Step Solution

Answer

The mathematical model that represents the statement 'A varies directly as the square of r' is \(A = k r^2\), where \(k\) is the constant of variation.

Step 1: Understand the Problem

The problem states that 'A' is directly proportional to the square of 'r'. Direct variation means that as one quantity increases, the other quantity increases by the same proportion, or conversely, as one decreases the other does too. The square of 'r' just means 'r' multplied by itself.

Step 2: Write the Variation Statement as an Equation

The statement 'A varies directly as the square of r' translates into an equation form as \(A = k r^2\), where \(k\) is a constant, also known as the constant of variation.

Step 3: Interpret the Equation

The equation \(A = k r^2\) means that 'A' is always equal to 'r' squared multiplied by a certain number, which equals 'A'. So if you know the values of 'r' and 'A', you can always find 'k', and this equation will hold true for any values of 'r' and 'k'.

Key Concepts

Understanding Proportionality

Proportionality is a fundamental mathematical concept that describes the relationship between two quantities where they change at the same rate. In a direct variation, as one quantity increases, the other quantity increases proportionally. This means they maintain a constant ratio. Direct variation is expressed in a simple form: when a variable is directly proportional to another, it can be expressed using the equation

• For example, if we say "A varies directly as the square of r," it implies a relation where 'A' and the square of 'r' increase in tandem, maintaining a constant ratio.
• This concept of maintaining a constant ratio is crucial and is represented by a constant, often denoted as 'k'.

Understanding proportionality helps in various real-life scenarios, such as estimating costs, predicting outcomes, and maintaining ratios across different values.

Exploring Mathematical Modeling

Mathematical modeling involves representing real-life situations through mathematical expressions, equations, or simulations. It is a powerful tool that helps in understanding complex relationships and solving problems effectively.

• In the context of direct variation, a mathematical model allows us to predict how changes in one variable will affect another variable.
• The purpose of creating a model is to simplify reality and provide actionable insights.

When modeling statements like 'A varies directly as the square of r,' the model formed is an equation that depicts the precise mathematical relationship. This is done by translating the verbal statement into a mathematical one:

• Here, the equation used is \(A = k r^2\). This serves as the mathematical model for the given scenario.
• This model can be used to calculate unknowns and understand dependencies. For example, predicting 'A' when 'r' is known, or determining the constant 'k'.

Mathematical modeling is pivotal in fields such as engineering, physical sciences, and economics, as it aids in making informed decisions.

Diving into Constant of Variation

The constant of variation is an integral component in the realm of direct variation. It signifies the fixed ratio that relates the two varying quantities in a direct variation equation.

• In the direct variation formula \(A = k r^2\), 'k' represents the constant of variation.
• This constant is vital because it retains the proportional relationship between 'A' and \(r^2\).

To find the constant of variation, you need values for 'A' and \(r\). By rearranging the formula:

• Divide 'A' by \(r^2\) to solve for 'k', giving: \(k = \frac{A}{r^2}\).
• Once 'k' is established, the relationship between any 'A' and 'r' can be determined using the equation. This helps maintain the consistency of the direct variation for any given pair of values.

The constant of variation is a concept applied in various situations, enabling reliable predictions and serving as a foundation for equations and further mathematical analysis.