Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as x and z. y = 25 when x = 2 and z = 5. Find y when x = 8 and z = 12.

Verified step by step guidance

1. Identify the type of variation described: since y varies jointly as x and z, we write the equation as \(y = k \cdot x \cdot z\), where \(k\) is the constant of proportionality.
2. Use the given values \(y = 25\), \(x = 2\), and \(z = 5\) to find the constant \(k\). Substitute these into the equation: \(25 = k \cdot 2 \cdot 5\).
3. Solve for \(k\) by isolating it on one side: \(k = \frac{25}{2 \cdot 5}\).
4. Write the general formula for \(y\) using the found value of \(k\): \(y = k \cdot x \cdot z\).
5. Substitute the new values \(x = 8\) and \(z = 12\) into the formula to find the new value of \(y\): \(y = k \cdot 8 \cdot 12\).

Key Concepts

Joint Variation

Joint variation describes a relationship where a variable depends on the product of two or more other variables. In this problem, y varies jointly as x and z means y = kxz, where k is a constant. Understanding this helps set up the correct equation to find unknown values.

Finding the Constant of Variation

To solve variation problems, first find the constant k by substituting known values of the variables into the joint variation equation. Here, using y = 25, x = 2, and z = 5 allows calculation of k, which is essential for determining y under new conditions.

Applying the Four-Step Procedure for Variation Problems

The four-step procedure typically involves identifying the variation type, writing the equation, finding the constant, and solving for the unknown. Following these steps systematically ensures accurate solutions in variation problems like this one.