Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(y\) is directly proportional to the square of \(x\) and inversely proportional to the cube of \(z\). If \(x=5\) and \(z=3\), then \(y=25\).

Step-by-Step Solution

Answer

The value of \( k \) is 27, and the formula is \( y = 27 \times \frac{x^2}{z^3} \).

Step 1: Understanding Direct and Inverse Proportions

The expression "directly proportional to the square of \(x\)" means \( y \propto x^2 \). Similarly, "inversely proportional to the cube of \(z\)" means \( y \propto \frac{1}{z^3} \). Since both conditions apply, we combine them to get \( y \propto \frac{x^2}{z^3} \).

Step 2: Introducing the Constant of Proportionality

To convert the proportional relationship into an equation, introduce a constant \(k\). The expression becomes \( y = k \left( \frac{x^2}{z^3} \right) \).

Step 3: Substituting Known Values

We need to find \(k\). Substitute the given values \(x=5\), \(z=3\), and \(y=25\) into the equation: \[ 25 = k \left( \frac{5^2}{3^3} \right) \]

Step 4: Solving for the Constant of Proportionality (k)

Calculate the expressions: \(5^2 = 25\) and \(3^3 = 27\). Plug these into the equation: \[ 25 = k \left( \frac{25}{27} \right) \]To solve for \(k\), multiply both sides by \(\frac{27}{25}\): \[ k = 25 \times \frac{27}{25} = 27 \]

Step 5: Final Formula with k

With \(k = 27\), the formula becomes \[ y = 27 \times \frac{x^2}{z^3} \].

Key Concepts

Proportionality Constant

The proportionality constant, denoted by the symbol \( k \), is a crucial element in relationships that involve direct or inverse proportion. It is the factor that turns a proportionality into an actual equation by providing a fixed multiplier. In this problem, for instance, \( y \) is influenced by the square of \( x \) and inversely by the cube of \( z \). To reflect this relationship effectively as a formula, we introduced \( k \).
By substituting known values of the variables into the equation, we can solve for \( k \). Once found, \( k \) gives us a precise and workable equation. This equation can be used to predict or identify \( y \) when other conditions vary. Understanding the role of the proportionality constant helps in interpreting the dynamics between dependent and independent variables.

Direct Variation

Direct variation describes a relationship where one variable increases or decreases with another variable in a manner that their ratio remains constant. When \( y \) varies directly with the square of \( x \), expressed as \( y \propto x^2 \), it means that as \( x \) increases, \( y \) will increase proportionally to \( x^2 \).
This type of variation makes it simpler to forecast changes in \( y \) based on changes in \( x \). It simplifies complex relationships by reducing them to simple multipliers.

• Remember, with direct variation, every change in \( x \) results in a predictable change in \( y \) based on the square function in this instance.
• Thus, the relationship can be verified easily with data points to ensure the variation factor remains consistent.

Inverse Variation

Inverse variation occurs when a variable decreases as the other variable increases, and their product is a constant. For example, when \( y \) is inversely proportional to the cube of \( z \), it is denoted by \( y \propto \frac{1}{z^3} \).
This signifies that as \( z \) grows, \( y \) diminishes in such a way that the multiplication of \( y \) and \( z^3 \) is consistent. Understanding inverse variation involves recognizing that such relationships are driven by dividing relationships instead of multiplying.

• In our exercise, combining inverse and direct variations helps paint a full picture of how \( y \) interacts with both \( x \) and \( z \).
• The phenomena offers a means to explain non-linear influences where an increase in a factor leads to a predictable decrease in the other.

Formula Derivation

Deriving the formula in exercises like this combines understanding both direct and inverse variations by interpreting and solving equations. Initially, the statement of the problem is represented proportionally as \( y \propto \frac{x^2}{z^3} \).
Introducing the constant \( k \) shifts the expression to an equation: \( y = k \frac{x^2}{z^3} \). From here, with known values provided, we calculate \( k \) to complete the derivation process.

• Substitute all given values into the derived equation. This allows you to isolate and solve for \( k \).
• Finally, substitute \( k \) back into the equation to produce the specific formula that now predicts \( y \).

The formula thus becomes a powerful tool to understand how changes in \( x \) and \( z \) relate to changes in \( y \) by directly incorporating their inverse and direct proportional impacts.