Problem 7 - College Algebra

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the cube of \(x\) and when \(x=36, y=24\).

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{1,944}x^3 \).

Step 1: Understand Direct Variation with Cube

In a variation problem where one variable varies directly as the cube of another variable, we can express this as \[ y = kx^3 \]where \(k\) is the constant of proportionality that needs to be determined.

Step 2: Substitute Known Values into the Equation

We are given that \(y = 24\) when \(x = 36\). Substitute these values into the equation\[ 24 = k(36)^3 \].

Step 3: Solve for the Constant of Proportionality \(k\)

Solve the equation from Step 2 to find \(k\).\[ 24 = k(36)^3 \]\[ 24 = k \times 46,656 \]Divide both sides by 46,656:\[ k = \frac{24}{46,656} \].

Step 4: Simplify and Calculate \(k\)

Simplify the fraction to find the value of \(k\). Divide 24 by 46,656:\[ k = \frac{1}{1,944} \].

Step 5: Write the Final Equation

Now that we have found \(k\), substitute it back into the equation \[ y = kx^3 \]. Thus, the relationship of the given variables is described by the equation: \[ y = \frac{1}{1,944}x^3 \].

Key Concepts

Constant of Proportionality

In mathematics, the constant of proportionality is the number that serves as a consistent multiplier in direct variation relationships. Direct variation happens when one variable changes, causing another variable to change in direct proportion. In our example, the relationship is between the variables \( y \) and \( x^3 \). We know from the problem that \( y \) varies directly as the cube of \( x \). Therefore, we can express this relationship with \( y = kx^3 \). Here, \( k \) is our constant of proportionality that ties the variation together.
By substituting the given values for \( x \) and \( y \), we calculate \( k \). Given \( y = 24 \) when \( x = 36 \), substituting into the equation \( y = kx^3 \), we solve \( k = \frac{24}{36^3} \). Finding \( k \) tells us how sensitive \( y \) is to changes in \( x \), specifically \( x^3 \), in this equation.

Cube of a Variable

A "cube of a variable" refers to raising that variable to the third power. In our exercise, it means using \( x \) such that \( x^3 \) is a significant component of our equation. Cubing a number multiplies the number by itself three times. So, if \( x = 36 \), then \( x^3 = 36 \times 36 \times 36 = 46,656 \).
This mathematical operation magnifies the values significantly, especially with larger numbers. Why is this important? Because when one variable is directly proportional to the cube of another, changes in that other variable have a considerably larger impact on the outcome variable compared to just a linear relationship. The cube not only makes the relationship unique but also more dynamic.

Equation Solving

Solving equations in the context of direct variation involves substituting known values and isolating the unknown. In our problem, the equation \( y = kx^3 \) was given, and we needed to find the value of \( k \).
We started by substituting the instance-specific values for \( y \) and \( x \) into the equation, yielding \( 24 = k(36)^3 \).

• Next, calculate \( 36^3 \), which is 46,656.
• Substitute back, giving \( 24 = 46,656k \).
• To solve for \( k \), divide both sides by 46,656, obtaining \( k = \frac{24}{46,656} \), which simplifies to \( \frac{1}{1,944} \).

This process of solving requires careful arithmetic and simplification to ensure accuracy in the variables' relationship equation \( y = \frac{1}{1,944}x^3 \). Equation solving like this helps in describing specific relationships clearly and correctly.