Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the cube root of \(x\). When \(x=27,\) then \(y=5 .\) Find \(y\) when \(x=125 .\)

Step-by-Step Solution

Answer

When \(x = 125\), then \(y = 3\).

Step 1: Understanding Inverse Variation

In inverse variation, a variable changes in such a way that as one variable increases, the other decreases. Mathematically, if \( y \) varies inversely with the cube root of \( x \), then \( y \) can be expressed as \( y = \frac{k}{\sqrt[3]{x}} \), where \( k \) is the constant of variation.

Step 2: Determine the Constant of Variation

Given \( y = 5 \) when \( x = 27 \), substitute these values into the inverse variation equation to find the constant \( k \).\[5 = \frac{k}{\sqrt[3]{27}}\]Since \( \sqrt[3]{27} = 3 \), this simplifies to:\[5 = \frac{k}{3} \implies k = 15\]

Step 3: Substitute the Known Values

Now that we have determined \( k = 15 \), use the equation \( y = \frac{15}{\sqrt[3]{x}} \) to find \( y \) when \( x = 125 \).\[y = \frac{15}{\sqrt[3]{125}}\]

Step 4: Calculate the Cube Root and Solve for y

Calculate the cube root of 125, which is 5, and substitute into the equation:\[y = \frac{15}{5} = 3\]

Step 5: Conclusion

Thus, when \( x = 125 \), the value of \( y \) is equal to 3.

Key Concepts

constant of variation

The constant of variation is a crucial component in the relationship between two variables. Particularly in inverse variation, this constant defines how two variables are inversely related. It's often symbolized by the letter \( k \). In our exercise, we saw that \( y \) varies inversely with the cube root of \( x \). This means as one value increases, the other decreases. The relationship is expressed as:\[ y = \frac{k}{\sqrt[3]{x}} \]To find the value of the constant \( k \), you need initial values of \( x \) and \( y \). Here, \( y = 5 \) when \( x = 27 \). By solving \( 5 = \frac{k}{3} \), we find that \( k = 15 \).

• This constant remains unchanged within the specific relationship or problem.
• It captures the intensity of the variation between \( y \) and \( x \).

Understanding the constant of variation helps in predicting how a change in \( x \) affects the value of \( y \) and vice versa.

cube root

The cube root is a mathematical operation that helps us determine a number which, when multiplied by itself three times, equals the original number. For instance, the cube root of 27 is 3, because \( 3 \times 3 \times 3 = 27 \). In the context of our problem, the variable \( x \) is used inside a cube root function.
The cube root is often denoted as \( \sqrt[3]{x} \). Here, the inverse variation of \( y \) is with the cube root of \( x \), which implies that:\[ y = \frac{k}{\sqrt[3]{x}} \]

• Cube roots are useful when dealing with volumes and three-dimensional shapes.
• They help in reducing the complexity when working with polynomials of degree three.

Being able to calculate the cube root makes solving such algebraic expressions easier and aids in understanding how variations work when they involve cubic terms.

algebraic expressions

Algebraic expressions are combinations of symbols and numbers. They can include variables, constants, coefficients, operators, and exponents. In solving problems involving inverse variation, like this one, algebraic expressions play a pivotal role. The specific form used here involves division and a root:\[ y = \frac{k}{\sqrt[3]{x}} \]This expression consists of:

• \( y \) as the dependent variable.
• \( k \) as the constant of variation.
• \( x \) under the cube root operation.

By substituting known values into the expression, we can solve for unknowns. For example, substituting \( x = 125 \) into the expression allowed us to solve for \( y \). Algebraic expressions facilitate calculations and express relationships clearly, revealing underlying patterns and making complex problems more understandable.