The maximum area of a traingle whose sides a,b,c satify `0 le a le 1,1 le b le 2,2 le c le 3 ` is

Options

A. 2
B. 1
C. `0.5`
D. none of these

Answer

To find the maximum area of a triangle with sides \( a, b, c \) satisfying the constraints \( 0 \leq a \leq 1 \), \( 1 \leq b \leq 2 \), and \( 2 \leq c \leq 3 \), we can follow these steps:

### Step 1: Understand the Area of a Triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times a \times b \times \sin C \]
where \( C \) is the angle opposite to side \( c \).

### Step 2: Identify the Constraints
From the problem, we have the following constraints for the sides of the triangle:
- \( 0 \leq a \leq 1 \)
- \( 1 \leq b \leq 2 \)
- \( 2 \leq c \leq 3 \)

### Step 3: Maximize the Area
To maximize the area, we need to maximize \( a \), \( b \), and \( \sin C \). The maximum value of \( a \) is 1, and the maximum value of \( b \) is 2. The sine function achieves its maximum value of 1 when \( C = 90^\circ \) (or \( \frac{\pi}{2} \) radians).

### Step 4: Calculate the Maximum Area
Substituting the maximum values into the area formula:
\[ A_{\text{max}} = \frac{1}{2} \times 1 \times 2 \times \sin\left(\frac{\pi}{2}\right) \]
Since \( \sin\left(\frac{\pi}{2}\right) = 1 \), we have:
\[ A_{\text{max}} = \frac{1}{2} \times 1 \times 2 \times 1 = 1 \]

### Conclusion
The maximum area of the triangle is \( \boxed{1} \).