Let's break this down step by step:

Given:

• $z$ varies jointly with the cube of $x$ and the square of $y$.
• This means we can express $z$ as: $z = k \cdot x^3 \cdot y^2$ where $k$ is the constant of proportionality.

Step 1: Find the constant of proportionality $k$

We are given:

• $z = 403.2$
• $x = 4$
• $y = 3$

Substitute these values into the variation equation: $403.2 = k \cdot 4^3 \cdot 3^2$

Now, calculate $4^3 = 64$ and $3^2 = 9$, so: $403.2 = k \cdot 64 \cdot 9$ $403.2 = k \cdot 576$ Solve for $k$: $k = \frac{403.2}{576} \approx 0.7$

So, $k \approx 0.7$.

Step 2: Write the variation equation

Now that we know $k$, the variation equation becomes: $z = 0.7 \cdot x^3 \cdot y^2$

Step 3: Find $z$ when $x = 9$ and $y = 9$

Using the equation: $z = 0.7 \cdot 9^3 \cdot 9^2$ First, calculate $9^3 = 729$ and $9^2 = 81$: $z = 0.7 \cdot 729 \cdot 81$ Now multiply: $z = 0.7 \cdot 59049 = 41334.3$

So, $z \approx 41334.3$.

Let me know if you need more details or explanations.

Questions for further practice:

1. What would $z$ be if $x = 2$ and $y = 5$?
2. How does changing the value of $x$ affect the value of $z$?
3. How would the variation equation change if $z$ varied directly with the square of $x$ instead of the cube?
4. Find the constant of proportionality if $z = 1200$, $x = 6$, and $y = 4$.
5. If the cube of $x$ doubled, by what factor would $z$ change?

Tip:

When solving variation problems, always substitute the known values carefully to avoid calculation errors, and double-check your exponent calculations.