Find an equation of variation where y varies jointly as x and the square of​ z, and wherenbsp y equals 66 when xequals15 and zequals2.

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.

Find an equation of variation where y varies jointly as x and the square of z, and where y equals 66 when x equals 15 and z equals 2.

Learn how to find an equation of variation where y varies jointly as x and the square of z. In this problem, y equals 66 when x equals 15 and z equals 2. The solution covers key algebraic concepts and joint variation formulas, suitable for students in Grades 9-12.

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