To express the relationship where $w$ varies jointly with $u$ and $d$, and inversely with $p$, we use the following form:

$w = k \cdot \frac{u \cdot d}{p}$

Here, $k$ is the constant of proportionality.

Explanation:

• Joint variation with $u$ and $d$: This means $w$ is directly proportional to the product of $u$ and $d$. This is represented as $w \propto u \cdot d$.
• Inverse variation with $p$: This means $w$ is inversely proportional to $p$, represented as $w \propto \frac{1}{p}$.

Combining these two variations, we get the equation:

$w = k \cdot \frac{u \cdot d}{p}$

Would you like further details or have any questions?

Related Questions:

1. What is the value of $k$ if $w = 20$, $u = 5$, $d = 4$, and $p = 2$?
2. How would the equation change if $w$ varied directly with $p$ instead of inversely?
3. If $u$ is doubled and $p$ is halved, by what factor will $w$ change?
4. What happens to $w$ if both $u$ and $d$ are tripled and $p$ is unchanged?
5. How can we use this relationship to solve practical problems in physics or engineering?

Tip:

When dealing with joint and inverse variation problems, always identify what the variables are directly or inversely related to in order to set up the correct proportionality.