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.

Verified step by step guidance

1. Start by writing the relationship given: since \( a \) is directly proportional to \( m \) and \( n^2 \), and inversely proportional to \( y^3 \), we can express this as \( a = k \cdot \frac{m \cdot n^2}{y^3} \), where \( k \) is the constant of proportionality.
2. Use the initial values \( a=9 \), \( m=4 \), \( n=9 \), and \( y=3 \) to find the constant \( k \). Substitute these into the equation: \( 9 = k \cdot \frac{4 \cdot 9^2}{3^3} \).
3. Simplify the expression inside the fraction: calculate \( 9^2 \) and \( 3^3 \), then solve for \( k \) by isolating it on one side of the equation.
4. Once you have \( k \), use it to find \( a \) when \( m=6 \), \( n=2 \), and \( y=5 \) by substituting these values into the formula: \( a = k \cdot \frac{6 \cdot 2^2}{5^3} \).
5. Simplify the expression by calculating \( 2^2 \) and \( 5^3 \), then multiply by \( k \) to find the new value of \( a \).

Key Concepts

Direct and Inverse Proportionality

Direct proportionality means one variable increases as another increases, expressed as a constant times the variables. Inverse proportionality means one variable increases as another decreases, represented by dividing by the variable. Understanding these relationships helps set up equations relating the variables.

Formulating Proportional Relationships

When a variable is directly proportional to some variables and inversely proportional to others, the relationship can be written as a product of the direct variables divided by the product of the inverse variables, multiplied by a constant. This formula allows solving for unknown values given initial conditions.

Solving for the Constant of Proportionality

To find the constant in a proportional relationship, substitute the known values of all variables into the equation. Solving for the constant enables calculation of the unknown variable when other variables change, making it essential for applying proportionality in problem-solving.