Problem 11 - 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 root of \(x\) and inversely proportional to the cube of \(z\). If \(x=9\) and \(z=2\), then \(y=5\).

Step-by-Step Solution

Answer

\( k = \frac{40}{3} \). The formula is \( y = \frac{40}{3} \cdot \frac{\sqrt{x}}{z^3} \).

Step 1: Set up the proportional relationship

Given that \(y\) is directly proportional to \(\sqrt{x}\) and inversely proportional to \(z^3\), we can express this relationship with a formula:\[ y = k \cdot \frac{\sqrt{x}}{z^3} \]where \(k\) is the constant of proportionality.

Step 2: Substitute the given values into the equation

We are given the condition that when \(x = 9\), \(z = 2\), and \(y = 5\). Substitute these values into the equation:\[ 5 = k \cdot \frac{\sqrt{9}}{2^3} \]Simplify further by calculating \(\sqrt{9}\) and \(2^3\):\[ 5 = k \cdot \frac{3}{8} \]

Step 3: Solve for the constant of proportionality

Now solve the equation for \(k\). Start by multiplying both sides by 8 to clear the fraction:\[ 40 = 3k \]Next, divide both sides by 3 to solve for \(k\):\[ k = \frac{40}{3} \]

Step 4: Write the complete formula with the calculated constant

Replace \(k\) in the original proportionality formula with its calculated value:\[ y = \frac{40}{3} \cdot \frac{\sqrt{x}}{z^3} \]This formula now completely describes the relationship between \(y\), \(x\), and \(z\) given the conditions.

Key Concepts

directly proportional

When two variables are directly proportional, it means that they increase or decrease in tandem. In simpler terms, as one variable goes up, the other goes up too, and vice versa.
Mathematically, we denote this relationship as: \( y \propto x \). This can be converted into an equation by introducing a constant of proportionality. Therefore, \( y = kx \), where \( k \) is a constant number that scales the relationship.
For the given problem, \( y \) is directly proportional to the square root of \( x \). So, if the square root of \( x \) increases, \( y \) will also increase accordingly. This aspect is vital because it defines how \( y \) scales with respect to changes in \( x \).

inversely proportional

If two quantities are inversely proportional, as one increases, the other decreases at a consistent rate. The relationship can be represented mathematically as \( y \propto \frac{1}{x} \). Similar to direct proportionality, it transforms into an equation by introducing a constant: \( y = \frac{k}{x} \).
In this exercise, \( y \) is inversely proportional to the cube of \( z \). This means as the value of \( z \) increases, the value of \( y \) decreases, keeping a balanced relationship. Recognizing inversely proportional relationships helps adjust one variable according to changes in the other.

constant of proportionality

A constant of proportionality is a fixed value that relates two or more variables in a proportional relationship. For both direct and inverse proportionality, this constant \( k \) bridges the gap to form precise equations.
In the problem, solving for \( k \) was crucial to complete the formula \( y = k \cdot \frac{\sqrt{x}}{z^3} \). The constant determines how much impact \( x \) and \( z \) have on \( y \). Finding \( k \) involved substituting the provided values into the equation and simplifying. This process of solving for \( k \) allows us to fully describe the relationship under given conditions.

mathematical formula

A mathematical formula is an expression that denotes a relationship between different quantities. In this scenario, the formula \( y = k \cdot \frac{\sqrt{x}}{z^3} \) encapsulates the relationship given the conditions.
Creating this formula involves understanding both direct and inverse proportionality, then combining them. It's a consolidated representation detailing how \( x \), \( z \), and \( y \) interact.
The formula allows us to predict the value of \( y \) when knowing \( x \) and \( z \), using \( k \) as the instrument that adjusts according to given conditions.