https://file.aiquickdraw.com/mathbot/file/1724193331130vkby2wyz.jpg

To solve the problem using the provided density function, let's follow these steps:

1. **Identify the relevant interval and height**: 
   - The density function is a uniform distribution over the interval [2, 7].
   - The height of the density function (which represents the probability density) is 0.2.

2. **Calculate the probability \( P(X < 4.52) \)**:
   - This is the area under the density function from \( X = 2 \) to \( X = 4.52 \).

3. **Compute the area**:
   - The base of the rectangle (interval length) is \( 4.52 - 2 = 2.52 \).
   - The height of the rectangle is given as 0.2.
   - Therefore, the probability is given by:
   \[
   P(X < 4.52) = \text{base} \times \text{height} = 2.52 \times 0.2 = 0.504
   \]

4. **Round the answer to two decimal places**:
   \[
   P(X < 4.52) \approx 0.50
   \]

So, the probability \( P(X < 4.52) \) is approximately **0.50**.

Would you like more details or have any further questions? Here are some related questions you might find useful:

1. What is the probability \( P(X > 5) \) given the same density function?
2. How would you calculate the probability \( P(3 < X < 6) \) using this density function?
3. What is the cumulative distribution function (CDF) for this uniform distribution?
4. How would the probability change if the interval for the density function was [1, 8]?
5. What is the expected value (mean) of this distribution?

**Tip:** The area under the curve of a probability density function (PDF) for continuous distributions represents the probability for a given interval.

Learn how to calculate the probability P(X < 4.52) using the uniform distribution density function with a step-by-step solution. This problem involves determining the area under the density function over a specified interval. Suitable for advanced high school and undergraduate students.

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