Problem 1 - Algebra for College Students

Question

\(y\) varies directly as the cube of \(x . y=k x^{3}\)

Step-by-Step Solution

Answer

\( y \) varies directly with the cube of \( x \) according to the equation \( y = kx^3 \).

Step 1: Understand the direct variation formula

In a direct variation, one variable varies directly as another variable raised to a power. Here, the formula given is \( y = kx^3 \), where \( k \) is the constant of variation, and \( y \) varies directly as the cube of \( x \). This means as \( x \) increases or decreases, \( y \) will increase or decrease proportionally to \( x^3 \).

Step 2: Identify the constant of variation

To solve for specific values of \( x \) or \( y \), we need the constant \( k \). If you have a pair of \( x \) and \( y \) values, substitute them into the equation \( y = kx^3 \) to find \( k \). For example, if \( x = a \) and \( y = b \), then \( b = k a^3 \), so \( k = \frac{b}{a^3} \).

Step 3: Use the equation to make predictions

If \( k \) is known, you can substitute any value of \( x \) into \( y = kx^3 \) to find the corresponding value of \( y \). Similarly, if you know \( y \) and need to find \( x \), rearrange the equation: \( x = \left(\frac{y}{k}\right)^{1/3} \).

Key Concepts

Cube of a Number

In mathematics, a cube of a number refers to multiplying the number by itself twice. For instance, if we take the number 3, its cube is calculated as follows:

• First, multiply the number by itself: \( 3 \times 3 = 9 \)
• Then, multiply the result by the original number again: \( 9 \times 3 = 27 \)

So, the cube of 3 is 27, which can also be written as \(3^3 = 27\). You'll find the cube of a number often noted in mathematical equations, especially in algebra, to express exponential growth rates. In the exercise solution, we have \( y = kx^3 \), which shows a relationship where \( y \) is directly proportional to the cube of \( x \). This means that small changes in \( x \) lead to intense variations in \( y \) because any change in \( x \) is magnified as a cube.

Constant of Variation

The constant of variation, denoted as \( k \), is a crucial part of direct variation equations. It helps us understand how one variable changes in relation to another. In our given equation, \( y = kx^3 \), \( k \) acts as the scaling factor that connects \( y \) and the cube of \( x \). For any direct variation scenario, the constant \( k \) determines how steep or gradual the changes in \( y \) occur as \( x \) changes. A larger \( k \) means that \( y \) changes more dramatically for a given change in \( x \), while a smaller \( k \) indicates a more subtle change. To find \( k \), you need specific pairs of \( x \) and \( y \), which you can insert into the equation \( y = kx^3 \). Solving for \( k \) using this formula requires rearranging it as \( k = \frac{y}{x^3} \), giving you the constant once you have values for \( x \) and \( y \). Knowing \( k \) allows you to predict how changes in \( x \) will affect \( y \).

Algebraic Equations

Algebraic equations are mathematical statements of equality comprising variables and constants. They often involve operations like addition, subtraction, multiplication, division, and exponentiation. In our context, \( y = kx^3 \) is an algebraic equation that models a direct variation trend.Understanding algebraic equations involves several steps:

• Identify the given variables and constants.
• Determine the mathematical operation connecting the variables.
• Use these relationships to solve for unknowns.

In direct variation within algebra, such as our equation, solving equations involves substituting known values into the equation and rearranging to find the unknowns. This represents a powerful way to relate variables, making predictions more accessible and reliable.