Two friends visit a restaurant randomly during 5 pm to 6 pm . Among the two, whoever comes first waits for 15 min and then leaves. The probability that they meet is :

Options

A. `(1)/(4)`
B. `(1)/(16)`
C. `(7)/(16)`
D. `(9)/(16)`

Answer

To find the probability that two friends meet at a restaurant given the conditions of their arrival, we can follow these steps:

### Step 1: Define the Problem
The two friends arrive randomly between 5 PM and 6 PM, which is a total time interval of 60 minutes. Let us denote the time of arrival of the first friend as \( x \) and the second friend as \( y \), where both \( x \) and \( y \) are uniformly distributed over the interval [0, 60].

### Step 2: Determine Meeting Conditions
The first person to arrive waits for 15 minutes. Therefore, for the two friends to meet, the following condition must hold:
- If friend \( x \) arrives first, then friend \( y \) must arrive within 15 minutes after \( x \).
- This can be expressed mathematically as:
\[ y \leq x + 15 \]
- Conversely, if friend \( y \) arrives first, then friend \( x \) must arrive within 15 minutes after \( y \):
\[ x \leq y + 15 \]

### Step 3: Set Up the Inequalities
Combining these conditions, we can express the meeting conditions as:
\[ -15 \leq x - y \leq 15 \]
This can be rewritten as two inequalities:
1. \( x - y \leq 15 \)
2. \( x - y \geq -15 \)

### Step 4: Graph the Inequalities
We can graph these inequalities on a coordinate plane where the x-axis represents \( x \) (arrival time of friend 1) and the y-axis represents \( y \) (arrival time of friend 2). The lines corresponding to the inequalities are:
- \( y = x - 15 \) (for \( x - y \leq 15 \))
- \( y = x + 15 \) (for \( x - y \geq -15 \))

### Step 5: Identify the Feasible Region
The feasible region where the two friends can meet is bounded by the lines \( y = x - 15 \) and \( y = x + 15 \) within the square defined by \( 0 \leq x \leq 60 \) and \( 0 \leq y \leq 60 \).

### Step 6: Calculate the Area of the Feasible Region
The total area of the square is:
\[ \text{Total Area} = 60 \times 60 = 3600 \]
The area of the triangle that does not meet the conditions (the area outside the meeting region) can be calculated. The vertices of the triangle formed by the lines and the axes are:
- \( (0, 15) \)
- \( (45, 60) \)
- \( (60, 45) \)

The area of this triangle can be calculated as:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 45 \times 45 = 1012.5 \]
Since there are two such triangles (one in each corner), the total area of the triangles is:
\[ \text{Total Area of Triangles} = 2 \times 1012.5 = 2025 \]

### Step 7: Calculate the Area of the Meeting Region
The area of the meeting region is:
\[ \text{Area of Meeting Region} = \text{Total Area} - \text{Total Area of Triangles} = 3600 - 2025 = 1575 \]

### Step 8: Calculate the Probability
The probability that they meet is given by the ratio of the area of the meeting region to the total area:
\[ P(\text{meeting}) = \frac{\text{Area of Meeting Region}}{\text{Total Area}} = \frac{1575}{3600} = \frac{7}{16} \]

### Final Answer
Thus, the probability that the two friends meet is \( \frac{7}{16} \).

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