Question

write a general formula to describe each variation. The cube of \(z\) varies directly with the sum of the squares of \(x\) and \(y, \quad z=2\) when \(x=9\) and \(y=4\)

Short answer

\[ z^3 = \frac{8}{97}(x^2 + y^2) \]

Step-by-step solution

Identify the Type of Variation

Determine the relationship between the variables. The problem states that the cube of \(z\) varies directly with the sum of the squares of \(x\) and \(y\), implying a direct variation.

Write the General Formula

Since \(z^3\) varies directly with \(x^2 + y^2\), we can express this relationship using a constant of proportionality \(k\). The formula will be: \[ z^3 = k(x^2 + y^2) \]

Substitute Given Values to Find \(k\)

Use the given values of \(z = 2\), \(x = 9\), and \(y = 4\) to find the constant \(k\). Substitute these values into the formula: \[ 2^3 = k(9^2 + 4^2) \] Simplify: \[ 8 = k(81 + 16) \] \[ 8 = k(97) \] Solve for \(k\): \[ k = \frac{8}{97} \]

Write the Final General Formula

Substitute the constant \(k\) back into the general formula: \[ z^3 = \frac{8}{97}(x^2 + y^2) \]

Key concepts

Proportionality Constant

A proportionality constant, often symbolized as \( k \), is a number that defines the relationship between two variables that vary directly. In direct variation, as one variable increases or decreases, the other does so proportionally. For instance, if \( y \) varies directly with \( x \), the relationship can be written as \( y = kx \). Here, \( k \) determines how much \( y \) changes for a given change in \( x \).

In the given exercise, we determined that the cube of \( z \) varies directly with the sum of the squares of \( x \) and \( y \). Hence, the general form is \( z^3 = k(x^2 + y^2) \). To find the specific value of \( k \), known values of \( x, y, \) and \( z \) are substituted into this equation.

Cube of a Variable

The cube of a variable, represented as \( z^3 \), means the variable \( z \) is multiplied by itself three times. For example, if \( z = 2 \), then \( z^3 = 2 \times 2 \times 2 = 8 \). Cubing a variable can significantly change its value, making it grow much faster than multiplying by itself just once. This relationship is essential in many areas of math and physics, as it illustrates exponential growth.

In the exercise, the cube of \( z \) is investigated, and we start with \( z = 2 \), leading us to \( 2^3 = 8 \). This is crucial for finding the proportionality constant and deriving the final formula.

Square of a Variable

The square of a variable, shown as \( x^2 \) or \( y^2 \), means the variable is multiplied by itself. For instance, if \( x = 9 \), then \( x^2 = 9 \times 9 = 81 \). Squaring causes numbers to increase very quickly, similar to cubing, making it important in describing relationships involving area, energy, and in our case, variations.

In the provided exercise, we utilized the squares of \( x \) and \( y \) to find the direct variation. For \( x = 9 \) and \( y = 4 \), we calculate \( x^2 = 81 \) and \( y^2 = 16 \). These values are then summed up, \( 81 + 16 = 97 \), and used in the formula \( z^3 = k(x^2 + y^2) \).