Area bounded by the relation [2x]+[y]=5, x,y `gt` 0 is (where `[*]` represent greatest integer function)

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the area bounded by the relation \([2x] + [y] = 5\) for \(x, y > 0\), we will analyze the equation step by step.

### Step 1: Understanding the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). Therefore, \([2x]\) can take values based on the range of \(x\).

### Step 2: Analyzing \([2x]\)
Let’s consider the possible integer values for \([2x]\):
- If \(0 \leq 2x < 1\), then \([2x] = 0\) which implies \(0 \leq x < \frac{1}{2}\).
- If \(1 \leq 2x < 2\), then \([2x] = 1\) which implies \(\frac{1}{2} \leq x < 1\).
- If \(2 \leq 2x < 3\), then \([2x] = 2\) which implies \(1 \leq x < \frac{3}{2}\).
- If \(3 \leq 2x < 4\), then \([2x] = 3\) which implies \(\frac{3}{2} \leq x < 2\).
- If \(4 \leq 2x < 5\), then \([2x] = 4\) which implies \(2 \leq x < \frac{5}{2}\).

### Step 3: Finding Corresponding \(y\) Values
For each value of \([2x]\), we can find the corresponding values of \([y]\) using the equation \([2x] + [y] = 5\):
- If \([2x] = 0\), then \([y] = 5\) which gives \(5 \leq y < 6\).
- If \([2x] = 1\), then \([y] = 4\) which gives \(4 \leq y < 5\).
- If \([2x] = 2\), then \([y] = 3\) which gives \(3 \leq y < 4\).
- If \([2x] = 3\), then \([y] = 2\) which gives \(2 \leq y < 3\).
- If \([2x] = 4\), then \([y] = 1\) which gives \(1 \leq y < 2\).

### Step 4: Plotting the Points
Now we can summarize the ranges:
1. For \(0 \leq x < \frac{1}{2}\), \(5 \leq y < 6\).
2. For \(\frac{1}{2} \leq x < 1\), \(4 \leq y < 5\).
3. For \(1 \leq x < \frac{3}{2}\), \(3 \leq y < 4\).
4. For \(\frac{3}{2} \leq x < 2\), \(2 \leq y < 3\).
5. For \(2 \leq x < \frac{5}{2}\), \(1 \leq y < 2\).

### Step 5: Calculating the Area
Each of these ranges forms a rectangle in the first quadrant:
- The area of the rectangle for \(0 \leq x < \frac{1}{2}\) and \(5 \leq y < 6\) is \(\frac{1}{2} \times 1 = \frac{1}{2}\).
- The area for \(\frac{1}{2} \leq x < 1\) and \(4 \leq y < 5\) is \(\frac{1}{2} \times 1 = \frac{1}{2}\).
- The area for \(1 \leq x < \frac{3}{2}\) and \(3 \leq y < 4\) is \(\frac{1}{2} \times 1 = \frac{1}{2}\).
- The area for \(\frac{3}{2} \leq x < 2\) and \(2 \leq y < 3\) is \(\frac{1}{2} \times 1 = \frac{1}{2}\).
- The area for \(2 \leq x < \frac{5}{2}\) and \(1 \leq y < 2\) is \(\frac{1}{2} \times 1 = \frac{1}{2}\).

### Total Area Calculation
Total area = \(\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{5}{2}\).

### Final Answer
The area bounded by the relation \([2x] + [y] = 5\) for \(x, y > 0\) is \( \frac{5}{2} \) square units.

---