To solve this problem, we use the formula for joint variation.

When $y$ varies jointly as $x$ and the square of $z$, the relationship can be written as:

$y = k \cdot x \cdot z^2$

Where:

• $k$ is the constant of proportionality,
• $x$ is one variable,
• $z$ is another variable, and
• $y$ is the dependent variable.

Step 1: Write the equation with the given values.

Given:

• $y = 66$,
• $x = 15$,
• $z = 2$.

Substitute these values into the equation:

$66 = k \cdot 15 \cdot (2)^2$

$66 = k \cdot 15 \cdot 4$

$66 = k \cdot 60$

Step 2: Solve for $k$.

To find $k$, divide both sides by 60:

$k = \frac{66}{60} = 1.1$

Step 3: Write the final equation.

Now that we know $k = 1.1$, substitute it back into the original equation:

$y = 1.1 \cdot x \cdot z^2$

This is the equation of variation where $y$ varies jointly as $x$ and the square of $z$.

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Do you have any questions or would you like further details?

Here are 5 related questions:

1. What is joint variation, and how does it differ from direct variation?
2. Can you derive the formula for inverse variation?
3. How would the equation change if $y$ varied inversely as $x$?
4. What would happen if $z$ were squared, but $x$ remained unaffected?
5. How can you calculate the constant of proportionality in different contexts?

Tip: In joint variation, keep an eye on how multiple variables influence each other to maintain their proportional relationships.