https://file.aiquickdraw.com/mathbot/file/1730679985095vjggkcyg.jpg

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?

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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.

Suppose z varies inversely with t and that z = 9 when t = 11. What is the value of z when t = 3?

Explore how to solve an inverse variation problem where z varies inversely with t. Given z = 9 when t = 11, learn how to find the value of z when t = 3 with step-by-step guidance. Suitable for students in Grades 8-10.

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