Solve each problem. Let a be directly proportional to m and n^2, and inversely proportional to y^3. If a=9when m=4, n=9, and y=3, find a when m=6, n=2, and y=5.

Key Concepts

Direct Proportionality

Direct proportionality means that one variable increases or decreases in direct relation to another variable. In this context, if 'a' is directly proportional to 'm' and 'n^2', it implies that as 'm' or 'n^2' increases, 'a' also increases, and vice versa. This relationship can be expressed mathematically as a = k * (m * n^2), where k is a constant.

Inverse Proportionality

Inverse proportionality indicates that as one variable increases, another variable decreases. Here, 'a' is inversely proportional to 'y^3', meaning that if 'y' increases, 'a' decreases. This relationship can be expressed as a = k' / (y^3), where k' is another constant. The combination of direct and inverse proportionality is crucial for solving the problem.

Constant of Proportionality

The constant of proportionality is a value that relates the variables in a proportional relationship. In this problem, we first determine the constant 'k' using the initial conditions provided (a=9, m=4, n=9, y=3). Once 'k' is found, it can be used to calculate 'a' for different values of 'm', 'n', and 'y' by substituting them into the derived formula, allowing us to find the new value of 'a'.