Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies directly as x and inversely as the square of z. y = 20 when x = 50 and z = 5. Find y when x = 3 and z = 6.

Verified step by step guidance

1. Identify the type of variation described: y varies directly as x and inversely as the square of z. This means the relationship can be written as \(y = k \frac{x}{z^2}\), where \(k\) is the constant of variation.
2. Use the given values \(y = 20\), \(x = 50\), and \(z = 5\) to find the constant \(k\). Substitute these into the equation: \(20 = k \frac{50}{5^2}\).
3. Simplify the expression inside the fraction: calculate \$5^2\( and then solve for \)k$ by isolating it on one side of the equation.
4. Once \(k\) is found, write the general formula for \(y\) again with the known \(k\): \(y = k \frac{x}{z^2}\).
5. Use the new values \(x = 3\) and \(z = 6\) in the formula to find the new value of \(y\). Substitute and simplify without calculating the final numeric answer.

Key Concepts

Direct and Inverse Variation

Direct variation means one variable increases as another increases, expressed as y = kx. Inverse variation means one variable decreases as another increases, such as y = k / z². Understanding how y varies directly with x and inversely with the square of z helps set up the correct equation.

Constant of Variation

The constant of variation (k) is a fixed value that relates variables in variation problems. It is found by substituting known values of variables into the variation equation. Once k is determined, it can be used to find unknown values of y for different x and z.

Four-Step Procedure for Solving Variation Problems

This procedure involves: 1) Writing the variation equation, 2) Finding the constant of variation using given values, 3) Substituting the constant back into the equation, and 4) Using the equation to find the unknown variable. Following these steps ensures a systematic solution.