The given problem asks to write an equation that expresses the relationship:

"w varies directly with the square of u" and using k as the constant of proportionality.

Direct Variation

In a direct variation, one quantity is directly proportional to another. If we say that $w$ varies directly with the square of $u$, this means:

$w = k u^2$

where:

• $w$ is the variable that depends on $u$,
• $u^2$ is the square of $u$,
• $k$ is the constant of proportionality.

Thus, the equation expressing the relationship is:

$w = k u^2$

Would you like further details or clarifications?

Here are 5 related questions to explore further:

1. How does changing the value of $k$ affect the relationship between $w$ and $u$?
2. If $u = 3$ and $k = 2$, what is the value of $w$?
3. What would happen if $w$ varied inversely with the square of $u$?
4. Can you find the value of $k$ if $w = 36$ and $u = 4$?
5. What would the equation look like if $w$ varied directly with the cube of $u$?

Tip:

Remember, in any direct variation problem, the constant $k$ can be found by dividing the dependent variable by the square (or cube, etc.) of the independent variable, once a specific point is known.