Problem 12 - Algebra and Trigonometry

Question

Write an equation that expresses the statement. A is proportional to the square of \(t\) and inversely proportional to the cube of \(x .\)

Step-by-Step Solution

Answer

The equation is \(A = k \cdot \frac{t^2}{x^3}\).

Step 1: Understand Proportionality

The statement says that A is proportional to the square of \(t\) and inversely proportional to the cube of \(x\). This means that as \(t\) increases, \(A\) will increase with the square of \(t\), and as \(x\) increases, \(A\) will decrease with the cube of \(x\).

Step 2: Use Proportional Relationship

To express this proportionality, we start by writing \(A\) with the proportional symbol: \(A \propto \frac{t^2}{x^3}\). This shows the direct relationship with \(t^2\) and inverse relationship with \(x^3\).

Step 3: Introduce the Constant of Proportionality

Replace the proportional symbol with an equal sign by introducing a constant of proportionality, \(k\). Thus, the equation becomes: \(A = k \cdot \frac{t^2}{x^3}\). This equation represents the given statement in mathematical terms.

Key Concepts

Proportionality

Proportionality is a fundamental concept in mathematics. It helps us describe situations where one quantity increases or decreases in relation to another. When we say quantity A is proportional to another quantity B, it means that a change in B will cause a corresponding change in A.

In simple terms, if B goes up, A will also go up in a predictable way. This pattern is captured by the proportionality symbol \( \propto \). For example, if A is proportional to the square of \( t \), we'd write \( A \propto t^2 \).

In this relationship, every time \( t \) doubles, \( A \) quadruples because it is a square effect. This proportionality allows us to predict how changes in \( t \) affect \( A \). Understanding this can be particularly useful in various scientific and engineering contexts where we work with complex relationships between variables.

Inverse Proportionality

Inverse proportionality is a concept that looks at how one quantity decreases as another increases. When A is inversely proportional to B, it means that as B gets larger, A gets smaller. Conversely, as B gets smaller, A increases.

This relationship is often seen in the natural world, such as in physics or chemistry. We can express inverse proportionality mathematically. For example, if A is inversely proportional to \( x^3 \), we can represent this as \( A \propto \frac{1}{x^3} \).

This expression tells us that as \( x \) increases, the cube of \( x \) also increases, causing A to decrease. It's a balancing act between the two quantities. Understanding inverse proportionality can aid in problem-solving. It helps predict behaviors in systems where actions have opposite reactions.

Constants of Proportionality

In mathematics, we introduce constants of proportionality to convert a proportionality into a precise equation. When A is proportional to some expression involving other variables, we say \( A = k \times \) that expression. Here, \( k \) is the constant of proportionality.

This constant stays the same as long as the relationship between the variables maintains its form. For instance, in the given exercise, the relationship is \( A = k \cdot \frac{t^2}{x^3} \). This tells us how A varies with \( t \) and \( x \), factoring in a constant value \( k \).

Why is \( k \) so important? Simply put, it customizes the general relationship to fit specific circumstances, like different units of measurement or environmental conditions. Calculating \( k \) from known values allows us to use this equation to predict outcomes for other variable values. This makes constants of proportionality crucial for detailed and accurate predictions.