Question

write an equation that expresses the statement. (use k as the constant of proportionality.) t is proportional to the product of the squares of n and w and inversely proportional to the cube of d.

Explanation:

Step1: Recall proportionality rules

If a quantity \( T \) is proportional to \( A \) and inversely proportional to \( B \), the relationship is \( T = k\frac{A}{B} \), where \( k \) is the constant of proportionality.

Step2: Identify \( A \) and \( B \)

Here, \( T \) is proportional to the product of the squares of \( N \) and \( w \), so \( A = N^2 \cdot w^2 \). And \( T \) is inversely proportional to the cube of \( d \), so \( B = d^3 \).

Step3: Form the equation

Substitute \( A \) and \( B \) into the proportionality formula: \( T = k\frac{N^2 w^2}{d^3} \).

Answer:

\( T = \frac{k N^2 w^2}{d^3} \)