Minimum vlaue of `|x-p| + |x-15| +|x-p-15|.` If `p le x le 15 and 0 lt p lt 15` :

Options

A. 30
B. 15
C. 10
D. 0

Answer

To find the minimum value of the expression \( |x - p| + |x - 15| + |x - p - 15| \) under the conditions \( p \leq x \leq 15 \) and \( 0 < p < 15 \), we will analyze the expression step by step.

### Step 1: Understand the Expression
The expression we need to minimize is:
\[ E(x) = |x - p| + |x - 15| + |x - (p + 15)| \]
Given the constraints \( p \leq x \leq 15 \), we will analyze the behavior of the expression within this interval.

### Step 2: Analyze Each Absolute Value
1. **For \( |x - p| \)**: Since \( x \geq p \), this simplifies to \( x - p \).
2. **For \( |x - 15| \)**: Since \( x \leq 15 \), this simplifies to \( 15 - x \).
3. **For \( |x - (p + 15)| \)**: Since \( p + 15 > 15 \) (because \( p < 15 \)), we have \( |x - (p + 15)| = (p + 15) - x \).

### Step 3: Substitute Simplified Absolute Values
Substituting these simplifications into the expression gives:
\[ E(x) = (x - p) + (15 - x) + ((p + 15) - x) \]

### Step 4: Simplify the Expression
Now, we simplify \( E(x) \):
\[ E(x) = x - p + 15 - x + p + 15 - x \]
Combining like terms:
\[ E(x) = 30 - x \]

### Step 5: Find the Minimum Value
To find the minimum value of \( E(x) = 30 - x \) under the constraint \( p \leq x \leq 15 \), we need to maximize \( x \) since \( E(x) \) decreases as \( x \) increases.

The maximum value of \( x \) in the interval is 15. Thus, we substitute \( x = 15 \):
\[ E(15) = 30 - 15 = 15 \]

### Conclusion
The minimum value of \( |x - p| + |x - 15| + |x - (p + 15)| \) for \( p \leq x \leq 15 \) and \( 0 < p < 15 \) is:
\[ \boxed{15} \]