Problem 93 - College Algebra with Modeling and Visualization

Question

Find the constant of proportionality \(k\) $$ y=k x^{3}, \text { and } y=64 \text { when } x=2 $$

Step-by-Step Solution

Answer

The constant of proportionality \( k \) is 8.

Step 1: Identify the Given Information

We know the equation is given as \( y = k x^3 \) and we have to find \( k \). The values provided are \( y = 64 \) when \( x = 2 \).

Step 2: Substitute the Given Values

Substitute \( y = 64 \) and \( x = 2 \) into the equation: \( 64 = k (2)^3 \).

Step 3: Simplify the Expression

Calculate \( 2^3 \), which is \( 8 \), so the equation becomes \( 64 = k \cdot 8 \).

Step 4: Solve for \( k \)

Divide both sides by \( 8 \) to isolate \( k \): \( k = \frac{64}{8} \).

Step 5: Calculate \( k \)

Perform the division: \( k = 8 \).

Key Concepts

Proportional relationships

Proportional relationships describe a connection between two quantities where their ratio remains constant. In other words, if one quantity changes, the other does so in a predictable way such that their division always yields the same number, called the "constant of proportionality."
Understanding this concept is crucial because it forms the foundation of many mathematical relationships and scenarios in real life. For example, if you know that the cost of apples is proportional to the weight you buy, you can use this constant ratio to find out how much you'll pay for any given weight.
In our exercise, this idea is represented with the equation \( y = k x^3 \). Here, \( k \) is the constant that links \( y \) and \( x^3 \), showing that no matter the value of \( x \), \( y \) will adjust accordingly as long as the ratio \( \frac{y}{x^3} = k \) stays the same. This kind of proportionality can also be extended to more complex expressions like cubic functions.

Algebraic expressions

Algebraic expressions consist of numbers, variables, and operators that represent a mathematical idea or relationships between quantities. They form the basis of algebra and are used to model real-world problems and solve equations.
In the exercise we're discussing, the algebraic expression given is \( y = k x^3 \). This expression tells us that \( y \) is dependent on the cube of \( x \), scaled by the constant \( k \).

• The variable \( y \) represents the output or dependent value.
• The variable \( x \) is the input or independent value, whose change affects \( y \).
• The constant \( k \) adjusts the relationship magnitude between \( y \) and \( x^3 \).

Algebraic expressions are essential because they allow us to set up equations and inequalities that can be manipulated to uncover unknown values, such as the constant \( k \) in this situation.

Solving equations

Solving equations is about finding the value of unknown variables that make the equation true. It involves manipulating mathematical expressions using operations such as addition, subtraction, multiplication, and division in order to isolate the variable in question.
In our example of finding the constant of proportionality \( k \), we solve the equation \( 64 = k \cdot 8 \), which resulted from substituting \( x = 2 \) into \( y = k x^3 \).
Here’s how that process unfolds:

• First, substitute given values into the equation. For \( x = 2 \), compute \( (2)^3 = 8 \).
• This substitution modifies the equation to \( 64 = k \cdot 8 \).
• Next, divide both sides by 8 to isolate \( k \), giving \( k = \frac{64}{8} \).
• Finally, complete the division, which simplifies to \( k = 8 \).

Solving equations allows us to unravel the relationships expressed in mathematical forms and determine exact values for variables, which in turn helps us understand and predict outcomes in various scenarios.