https://file.aiquickdraw.com/mathbot/file/1726262721623ymrlqyek.jpg

Let's break down the problem.

### Part a) Finding the height of the function, \( f(x) \)

The function \( f(x) \) represents a uniform probability distribution. For a uniform distribution, the total area under the function should equal 1 because the total probability must be 1.

From the graph:
- The function has a constant height over the interval \( [0, 12] \), which is the base of the rectangle.
- The width of the rectangle is 12 (from \( x = 0 \) to \( x = 12 \)).

To find the height \( f(x) \), we use the fact that the area of the rectangle must equal 1:
\[
\text{Area} = \text{height} \times \text{width}
\]
Since the total probability (area) is 1:
\[
\text{height} \times 12 = 1
\]
Solving for the height:
\[
\text{height} = \frac{1}{12}
\]

Thus, the height of the function \( f(x) \) is \( \frac{1}{12} \).

---

### Part b) Probability of the shaded region

The shaded region corresponds to the interval \( [9, 12] \) on the \( x \)-axis. The probability of an interval in a uniform distribution is proportional to the area under the function over that interval.

To find this probability, we calculate the area of the shaded rectangle:
\[
\text{Probability} = \text{height} \times \text{width of the shaded region}
\]
The height is \( \frac{1}{12} \), and the width of the shaded region (from 9 to 12) is \( 12 - 9 = 3 \). Thus, the probability is:
\[
\frac{1}{12} \times 3 = \frac{3}{12} = \frac{1}{4}
\]

So, the probability of the shaded region is \( \frac{1}{4} \).

---

### Summary of Answers:
a) The height of the function \( f(x) \) is \( \frac{1}{12} \).  
b) The probability of the shaded region is \( \frac{1}{4} \).

Would you like any further details or clarification?

Here are five related questions to further expand on this:
1. How would you find the probability for a different interval, say from \( x = 5 \) to \( x = 10 \)?
2. What happens to the height of \( f(x) \) if the interval changes to \( [0, 15] \)?
3. Can this uniform distribution model be applied to any other real-world scenarios? 
4. How would you interpret the cumulative distribution function (CDF) for this uniform distribution?
5. What is the expected value \( E(x) \) for this uniform distribution?

**Tip:** When dealing with probability distributions, always ensure the total area under the function equals 1, as this represents the total probability.

Given the uniform distribution with the base between x = 0 and x = 12, find a) the height of the function f(x) and b) the probability of the shaded region between x = 9 and x = 12. Provide answers as reduced fractions.

Learn how to solve a uniform distribution problem where the height of the function and the probability of a shaded region are calculated. Explore probability concepts for students in Grades 11-12.

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