Problem 141 - College Algebra with Modeling and Visualization

Question

The variable \(z\) varies jointly as the square root of \(x\) and the cube root of \(y .\) If \(z=10.8\) when \(x=4\) and \(y=8\) find \(z\) when \(x=16\) and \(y=27\).

Step-by-Step Solution

Answer

When \(x=16\) and \(y=27\), \(z=32.4\).

Step 1: Understand the problem statement

We are given that the variable \(z\) varies jointly as the square root of \(x\) and the cube root of \(y\). This mathematical relationship can be expressed as \(z = k \sqrt{x} \cdot \sqrt[3]{y}\), where \(k\) is a constant of proportionality.

Step 2: Substitute known values to find the constant

Use the given values \(z = 10.8\), \(x = 4\), and \(y = 8\) to find the constant \(k\). Substitute into the equation:\[ 10.8 = k \sqrt{4} \cdot \sqrt[3]{8} \]Calculate the square root and cube root:\(\sqrt{4} = 2\) and \(\sqrt[3]{8} = 2\), so the equation becomes:\[ 10.8 = k \cdot 2 \cdot 2 \]\[ 10.8 = 4k \]Solve for \(k\):\[ k = \frac{10.8}{4} = 2.7 \]

Step 3: Use the constant to find new value of \(z\)

Now that we have \(k = 2.7\), we can find \(z\) for new values \(x = 16\) and \(y = 27\). Substitute into the equation:\[ z = 2.7 \sqrt{16} \cdot \sqrt[3]{27} \]Calculate the square root and cube root:\(\sqrt{16} = 4\) and \(\sqrt[3]{27} = 3\), so the equation becomes:\[ z = 2.7 \cdot 4 \cdot 3 \]\[ z = 32.4 \]

Key Concepts

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4, because 4 multiplied by 4 equals 16. Square roots are often used in various mathematical relationships and equations. Knowing how to calculate a square root can help you solve joint variation problems. Let's break it down further:

• To calculate the square root, you generally use the square root symbol \(\sqrt{}\).
• The square root of 4 is 2, because \(2 \times 2 = 4\).
• When dealing with equations, substitute the square root value to simplify expressions.

In joint variation problems, understanding square roots helps you decode part of the relationship between variables. Knowing how to find the square root of a number makes solving problems like the one in the exercise smoother and more intuitive.

Exploring Cube Roots

A cube root of a number is different from a square root. It's the value that when used in multiplication three times, results in the original number. For instance, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\). Cube roots are useful when dealing with joint variation equations involving three dimensions.

• The cube root is symbolized by \(\sqrt[3]{}\).
• For 8, the cube root is 2, due to \(2 \times 2 \times 2 = 8\).
• Substituting the cube root simplifies expressions in equations.

Learning how to find cube roots allows solving equations with joint variation between or among variables efficiently. It is a key skill required to grasp the relationship depicted in the exercise, as it covers variable dependencies on the cube root.

The Proportionality Constant Simplified

Understanding the concept of the proportionality constant is crucial in solving joint variation problems. The proportionality constant, denoted as \(k\), links the variables together in an equation. This constant helps maintain the unique relationship between the involved variables.

• The equation describes how \(z\) changes with both \(\sqrt{x}\) and \(\sqrt[3]{y}\).
• The constant \(k\) is determined by inserting known values into the joint variation equation.
• For the exercise, once \(z\), \(x\), and \(y\) were known, we solved for \(k\) using \[k = \frac{z}{\sqrt{x} \cdot \sqrt[3]{y}}\].
• This \(k\) value then helps predict or understand how changes in \(x\) and \(y\) will affect \(z\).

The proportionality constant offers predictability and helps relate the variables in joint variation, making it a cornerstone in understanding such mathematical scenarios.