Problem 23 - Intermediate Algebra

Question

Find the constant of variation for each of the stated conditions. \(y\) is directly proportional to \(x\) and inversely proportional to the square of \(z\), and \(y=81\) when \(x=36\) and \(z=2\).

Step-by-Step Solution

Answer

The constant of variation is 9.

Step 1: Write the Relationship Equation

Construct the equation that represents the relationship. Since \(y\) is directly proportional to \(x\) and inversely proportional to the square of \(z\), the equation is: \[ y = k \cdot \frac{x}{z^2} \] where \(k\) is the constant of variation.

Step 2: Substitute Given Values

Substitute \(y = 81\), \(x = 36\), and \(z = 2\) into the relationship equation:\[ 81 = k \cdot \frac{36}{2^2} \] This simplifies to \[ 81 = k \cdot \frac{36}{4} \].

Step 3: Simplify the Fraction

Calculate \(\frac{36}{4}\):\[ \frac{36}{4} = 9 \]Substitute back into the equation to get: \[ 81 = k \cdot 9 \].

Step 4: Solve for the Constant of Variation

To find \(k\), divide both sides by 9:\[ k = \frac{81}{9} \]Calculate the result:\[ k = 9 \].

Key Concepts

Direct Variation

Direct variation describes a relationship between two variables where if one variable increases, the other does too, maintaining a constant rate of change. This relationship can be described with the formula:

• \( y = kx \)

where \( y \) and \( x \) are the variables, and \( k \) is the constant of variation.
For example, if \( y \) doubles every time \( x \) does, then they are in direct variation.
In the given exercise, \( y \) is directly proportional to \( x \), meaning as \( x \) increases, \( y \) increases in a linear fashion based on the constant \( k \). Understanding direct variation helps in solving problems where we need to either estimate or calculate unknown values when one value changes.

Inverse Variation

Inverse variation occurs when an increase in one variable leads to a proportional decrease in another, and vice versa. The formula representing this relationship is:

• \( y = \frac{k}{x} \)

Here, \( y \) and \( x \) are variables, and \( k \) is the constant of variation.
This means that as the denominator variable \( x \) gets larger, the result \( y \) becomes smaller.
In the exercise, \( y \) is inversely proportional to the square of \( z \), shown by the expression \( \frac{x}{z^2} \). As \( z \) grows, the denominator increases, reducing \( y \). Grasping the concept of inverse variation is crucial for situations where balancing or inverse relationships are involved, such as pressure and volume in physics (Boyle's Law).

Proportion

Proportion refers to the equation that signifies two ratios are equivalents to each other. Think of it as a comparison between two quantities, often expressed as

• \( \frac{a}{b} = \frac{c}{d} \)

Proportions are integral in solving real-world problems, such as calculating unknown heights or distances using triangles' similarity rules.
In the given problem, the concept of proportion is used implicitly to find the constant of variation.
The variables \( y \), \( x \), and \( z \) maintain a proportional relationship expressed by \( y = k \cdot \frac{x}{z^2} \). Thus, finding a proportional value, like the constant \( k \), becomes straightforward with proportion principles.