Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information to find the unknown value. \(y\) varies directly as the cube root of \(x\). When \(x=125,\) then \(y=15\). Find \(y\) when \(x=1,000\).

Step-by-Step Solution

Answer

\(y = 30\) when \(x = 1,000\).

Step 1: Understand Variation

Since \(y\) varies directly as the cube root of \(x\), we can say \(y = k \cdot \sqrt[3]{x}\), where \(k\) is the constant of proportionality.

Step 2: Find Constant of Proportionality

We need to find \(k\) first. Given that \(y = 15\) when \(x = 125\), substitute into the equation: \(15 = k \cdot \sqrt[3]{125}\). Since \(\sqrt[3]{125} = 5\), we have \(15 = k \cdot 5\). Hence, \(k = \frac{15}{5} = 3\).

Step 3: Use Equation to Find Unknown Value

Now that we have \(k = 3\), use the equation \(y = 3 \cdot \sqrt[3]{x}\) to find \(y\) when \(x = 1000\).

Step 4: Calculate the Cube Root

Calculate the cube root of \(x\) when \(x = 1000\): \(\sqrt[3]{1000} = 10\).

Step 5: Calculate New Value of y

Substitute \(\sqrt[3]{1000} = 10\) into the equation: \(y = 3 \cdot 10 = 30\).

Key Concepts

Cube Root Function

A cube root function is a type of function where the variable inside the function is raised to the power of one-third. This is denoted as \( \sqrt[3]{x} \). Cube roots are important for solving problems where you need to "undo" cube powers or find the side length of cubes given a volume.
They follow the inverse operation of cubing numbers; for instance, the cube root of 8 is 2 because \(2^3 = 8\). This function helps in equations where you need to find variables that involve cubed numbers.
When using cube roots in direct variation problems, you're often transforming a material measure, like volume, into another form, such as length. This relationship is described by the equation \( y = k \cdot \sqrt[3]{x} \), where \( k \) is a constant that influences the direct proportion.

Proportionality Constant

The proportionality constant, often denoted as \( k \), is a crucial element in direct variation equations. It indicates how \( y \) scales with \( x \). In direct variation, if \( y \) is directly proportional to some function of \( x \), like its cube root, you write \( y = k \cdot \sqrt[3]{x} \).
Here, \( k \) determines the relationship's strength and direction. For the example, knowing \( y = 15 \) when \( x = 125 \) helped find \( k \). Substituting these into the equation gives \( 15 = k \cdot 5 \), solving which provides \( k = 3 \).
Understanding \( k \) gives insights into how y changes with x. It shows that if x increases or decreases, y will change by a factor of \( k \) in the same direction, based solely on the value of x's cube root.

Solve Equations

Solving equations, particularly with direct variation, involves substituting known values into the equation to find unknowns. The standard form \( y = k \cdot \sqrt[3]{x} \) makes it easier to calculate y when x changes.
For example, we first solved for the constant \( k \) using given x and y values. Once \( k = 3 \) was found, solving for different x was straightforward. When \( x = 1000 \), the cube root is \( \sqrt[3]{1000} = 10 \).
This final calculation with \( y = 3 \cdot 10 \) yields \( y = 30 \). Keeping organized steps is helpful, first calculating the cube root and then using it to solve the equation. Practicing these steps builds confidence in tackling similar problems.