To solve this problem, let's break down the concept of inverse variation and apply it step-by-step.

Step 1: Understanding Inverse Variation

When $z$ varies inversely with $t$, it means: $z = \frac{k}{t}$ where $k$ is a constant.

Step 2: Finding the Constant $k$

We are given that $z = 9$ when $t = 11$. Substitute these values into the equation to find $k$: $9 = \frac{k}{11}$ Multiplying both sides by 11 to isolate $k$: $k = 9 \times 11 = 99$

Step 3: Using the Constant to Find $z$ When $t = 3$

Now, we know $k = 99$, so we can use the inverse variation formula again: $z = \frac{99}{t}$ Substitute $t = 3$: $z = \frac{99}{3} = 33$

Final Answer

$z = 33$

Would you like a further explanation on inverse variation, or do you have any questions?

Here are 5 related questions to explore further:

1. What would the value of $z$ be if $t = 5$?
2. How would the equation change if $z$ varied directly with $t$ instead of inversely?
3. If $z = 33$ when $t = 3$, what would $t$ be when $z = 22$?
4. How can we determine if a relationship is inverse or direct variation?
5. What happens to the value of $z$ as $t$ approaches infinity?

Tip: In inverse variation, as one variable increases, the other decreases. This property is helpful in identifying inverse relationships.