Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the cube root of \(x\) and when \(x=27, \quad y=15\).

Step-by-Step Solution

Answer

The equation is \( y = 5 \cdot \sqrt[3]{x} \).

Step 1: Understand the Direct Variation Relationship

When a variable varies directly as another, it means that there is a constant multiplier (k) between them. In this case, y varies directly as the cube root of x, which can be written as \( y = k \cdot \sqrt[3]{x} \).

Step 2: Substitute the Given Values

We know that when \(x = 27\), \(y = 15\). Substitute these values into the equation from Step 1: \(15 = k \cdot \sqrt[3]{27}\).

Step 3: Solve for the Constant of Variation (k)

First, calculate the cube root of 27. \( \sqrt[3]{27} = 3\). Substitute this back into the equation: \(15 = k \cdot 3\). Divide both sides by 3 to find \(k\): \(k = \frac{15}{3} = 5\).

Step 4: Write the Equation

Now that we know \(k = 5\), we can write the equation representing the relationship between y and the cube root of x: \( y = 5 \cdot \sqrt[3]{x} \).

Key Concepts

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because when 3 is cubed (or multiplied by itself twice), it results in 27: \[3 \times 3 \times 3 = 27\]Cube roots are especially useful in equations involving volume and direct variations.
To find the cube root of a number, you can use methods like:

• Prime factorization, if the number is simple or a perfect cube.
• Calculator or computational tool for more complex numbers.

In our example, the cube root of 27 was found to be 3, which was crucial in resolving the relationship between variables in the provided exercise.

Constant of Variation

In direct variation, the constant of variation is the specific constant \(k\) that links two variables in a linear relationship. It remains unchanged as long as the variation conditions are met. In our exercise, \(y\) varies directly with the cube root of \(x\). This situation can be described with the equation:\[y = k \cdot \sqrt[3]{x}\]Here, \(k\) is the constant of variation. It refers to how much \(y\) changes as the cube root of \(x\) changes. Finding \(k\) involves substituting known values into the equation.
For example:

• With \(x = 27\) and \(y = 15\), substitute these values into the variation equation.
• Compute \(\sqrt[3]{27} = 3\).
• Substitute back into the equation to find \(k\): \(15 = k \cdot 3\).
• Solving for \(k\), we conclude \(k = 5\).

This constant helps us understand the proportional changes between the variables across different values of \(x\) and \(y\).

Mathematical Equation

Mathematical equations are like bridges connecting different mathematical concepts. They allow us to describe relationships between variables using constants and operations. In direct variation problems, such an equation helps us see how one variable changes in response to another.
In our exercise, the relationship between \(y\) and the cube root of \(x\) is expressed as:\[y = k \cdot \sqrt[3]{x}\]This equation shows a direct proportion between \(y\) and the cube root of \(x\). By solving such equations, we can predict outcomes, calculate unknowns, and describe mathematical phenomena.
Consider when:

• \(x = 64\), we can calculate \(y\)
• The cube root of 64 is 4: \(\sqrt[3]{64} = 4\)
• Substitute into the equation with \(k = 5\): \(y = 5 \cdot 4 = 20\)

Understanding these relationships through equations allows students to approach problems systematically, using logic and reasoning.