Solve each problem. Let a be directly proportional to m and n², and inversely proportional to y³. 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. Identify the relationship given: \( a \) is directly proportional to \( m \) and \( n^2 \), and inversely proportional to \( y^3 \). This can be written as the equation \( a = k \cdot \frac{m \cdot n^2}{y^3} \), where \( k \) is the constant of proportionality.
2. Use the given values \( a=9 \), \( m=4 \), \( n=9 \), and \( y=3 \) to find the constant \( k \). Substitute these values into the equation: \( 9 = k \cdot \frac{4 \cdot 9^2}{3^3} \).
3. Simplify the expression on the right side to solve for \( k \). Calculate \( 9^2 \) and \( 3^3 \), then isolate \( k \) by dividing both sides of the equation by the resulting fraction.
4. Once \( k \) is found, use it to find \( a \) when \( m=6 \), \( n=2 \), and \( y=5 \). Substitute these values and \( k \) into the original formula: \( a = k \cdot \frac{6 \cdot 2^2}{5^3} \).
5. Simplify the expression on the right side to find the new value of \( a \). Calculate \( 2^2 \) and \( 5^3 \), then multiply and divide accordingly to express \( a \) in terms of \( k \).

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 Proportionality Equations

When a quantity is directly proportional to some variables and inversely proportional to others, it can be expressed as a product of the direct variables multiplied by a constant, divided by the product of the inverse variables raised to their powers. This forms the basis for solving for unknowns.

Solving for the Constant of Proportionality

To find the constant of proportionality, substitute the known values of all variables into the proportionality equation and solve for the constant. This constant is then used to find the unknown variable when other values change.