Problem 12 - 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 square root of \(z\). If \(x=5\) and \(z=16\), then \(y=10\).

Step-by-Step Solution

Answer

The formula is \( y = \frac{8}{5} \frac{x^2}{\sqrt{z}} \) and \( k = \frac{8}{5} \).

Step 1: Understand the Proportionality Relationship

The statement 'y is directly proportional to the square of x and inversely proportional to the square root of z' can be expressed in a formula involving a constant of proportionality, \( k \): \( y = k \frac{x^2}{\sqrt{z}} \). This means that \( y \) increases with the square of \( x \) and decreases as the square root of \( z \) increases.

Step 2: Substitute Given Values into the Formula

We know from the problem that \( x = 5 \), \( z = 16 \), and \( y = 10 \). Substitute these values into the proportional formula: \( 10 = k \frac{5^2}{\sqrt{16}} \).

Step 3: Simplify the Equation

Calculate the values: \( 5^2 = 25 \) and \( \sqrt{16} = 4 \). Substitute into the equation to get \( 10 = k \frac{25}{4} \).

Step 4: Solve for Constant of Proportionality (k)

To find \( k \), solve the equation: \( 10 = \frac{25}{4}k \). Rearrange to find \( k \): \( k = \frac{10 \times 4}{25} \). Simplifying gives \( k = \frac{40}{25} = \frac{8}{5} \).

Step 5: Verify the Calculations

It's good practice to double-check the calculations for correctness. Substituting \( k = \frac{8}{5} \) back into our original equation verifies that all values satisfy the equation correctly for the given \( x = 5 \), \( z = 16 \), and \( y = 10 \).

Key Concepts

Constant of Proportionality

The constant of proportionality, often symbolized as "\(k\)", is a crucial element in expressing direct and inverse proportionality relationships. It acts as a multiplier in formulas that ties variables together. When two variables are said to be proportional, it means they have a consistent ratio. This ratio is maintained across different values of these variables. For instance, in direct proportionality, one variable increases as the other increases, maintaining the same ratio. Conversely, in inverse proportionality, as one variable increases, the other decreases, but again, the ratio remains constant.

In our exercise, the statement is that \(y\) is directly proportional to the square of \(x\) and inversely proportional to the square root of \(z\). This is expressed by the mathematical relationship \(y = k \frac{x^2}{\sqrt{z}}\). This equation encapsulates both types of proportionality using the constant \(k\). Therefore, understanding \(k\) is key to solving such problems as it quantifies how strong the relationship between these variables is.

To determine the actual value of \(k\), specific data or conditions must be provided, and this requires calculating \(k\) by substituting known values into the equation.

Inverse Proportionality

Inverse proportionality describes a situation where one variable increases as the other variable decreases. Mathematically, this relationship can be expressed as \( y \propto \frac{1}{x} \). This indicates that \( y \) is proportional to the inverse of \( x \). In our example, \( y \) is inversely proportional to the square root of \( z \), which is shown in the equation as \( y = k \frac{x^2}{\sqrt{z}} \).

Here, if \( z \) increases, the term \( \frac{1}{\sqrt{z}} \) decreases, thus reducing \( y \). Conversely, decreasing \( z \) results in an increase in \( y \). The inverse relationship is essential in many scientific and engineering fields, such as physics, where the inverse square law is a common principle.

This kind of proportionality demands careful attention because it often leads to situations where changes in one variable have a significant impact on another, but in the opposite direction. It’s important to understand and interpret this relationship correctly, especially in scenarios like these where inversely proportional elements affect a directly proportional component.

Square and Square Root Functions

Square and square root functions play a crucial role in many proportionality problems, and they describe how a variable can impact another variable in a non-linear fashion. In our previous exercise, \( x^2 \) signifies that \( y \) varies with the square of \( x \). It means that if \( x \) is doubled, then the squared term \( x^2 \) will increase four times, thus having a quadrupled effect on \( y \).

On the other hand, the square root of \( z \), \( \sqrt{z} \), demonstrates how \( y \) changes inversely. Unlike square functions, which increase rapidly, square root functions grow more slowly. For example, if \( z \) quadruples, the \( \sqrt{z} \) will double. This slower rate of change can modulate or mitigate the direct impact of changes from \( x^2 \) to \( y \).

Understanding these functions is fundamental to mastering algebra and other higher-level mathematics, as they frequently appear in equations defining physical laws and geometric principles. Grasping how squares and square roots affect proportional relationships is crucial for effectively interpreting and solving mathematical problems.