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.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is a constant multiple of another. In mathematical terms, if y varies directly as x, it can be expressed as y = kx, where k is a non-zero constant. This concept is essential for understanding how changes in one variable affect another in a proportional manner.

Inverse Variation

Inverse variation occurs when one variable increases while another decreases, maintaining a constant product. If y varies inversely as the square of z, it can be represented as y = k/z². This relationship is crucial for solving problems where one variable's increase leads to the decrease of another, particularly in the context of the given problem.

Combined Variation

Combined variation involves both direct and inverse relationships among variables. In this case, y varies directly with x and inversely with the square of z, leading to the equation y = kx/z². Understanding combined variation is vital for solving complex problems where multiple variables interact in different ways, as seen in the exercise.