The maximum area of the convex polyon formed by joining the points `A(0,0),B(2t^2,0),C(18,2),D((8)/(r^2),4)` and `E(0,2)` where `tinR-{0}` and interior angle at vertex B is greater than or equal to `90^@` is

Explanation

To find the maximum area of the convex polygon formed by the points \( A(0,0), B(2t^2,0), C(18,2), D\left(\frac{8}{r^2},4\right), E(0,2) \) with the condition that the interior angle at vertex \( B \) is greater than or equal to \( 90^\circ \), we can follow these steps:

### Step 1: Identify the points and their coordinates
We have the following points:
- \( A(0,0) \)
- \( B(2t^2, 0) \)
- \( C(18, 2) \)
- \( D\left(\frac{8}{r^2}, 4\right) \)
- \( E(0, 2) \)

### Step 2: Determine the area of the polygon
The area of a polygon can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 + x_5y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_5 + y_5x_1) \right| \]
Substituting the coordinates of points \( A, B, C, D, E \):
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (2t^2, 0) \)
- \( (x_3, y_3) = (18, 2) \)
- \( (x_4, y_4) = \left(\frac{8}{r^2}, 4\right) \)
- \( (x_5, y_5) = (0, 2) \)

### Step 3: Calculate the area using the coordinates
Substituting into the area formula:
\[ \text{Area} = \frac{1}{2} \left| 0 \cdot 0 + 2t^2 \cdot 2 + 18 \cdot 4 + \frac{8}{r^2} \cdot 2 + 0 \cdot 0 - (0 \cdot 2t^2 + 0 \cdot 18 + 2 \cdot \frac{8}{r^2} + 4 \cdot 0 + 2 \cdot 0) \right| \]
This simplifies to:
\[ \text{Area} = \frac{1}{2} \left| 0 + 4t^2 + 72 + \frac{16}{r^2} + 0 - (0 + 0 + \frac{16}{r^2} + 0 + 0) \right| \]
\[ = \frac{1}{2} \left| 4t^2 + 72 \right| \]
\[ = 2t^2 + 36 \]

### Step 4: Maximize the area under the given conditions
To maximize the area \( 2t^2 + 36 \), we can observe that as \( t \) increases, the area increases. However, we need to ensure that the angle at \( B \) is \( \geq 90^\circ \).

### Step 5: Condition for angle at \( B \)
The angle at \( B \) will be \( \geq 90^\circ \) if the slopes of lines \( AB \) and \( BC \) satisfy:
\[ \text{slope of } AB \cdot \text{slope of } BC \leq -1 \]
Calculating the slopes:
- Slope of \( AB = \frac{0 - 0}{2t^2 - 0} = 0 \)
- Slope of \( BC = \frac{2 - 0}{18 - 2t^2} = \frac{2}{18 - 2t^2} \)

Since the slope of \( AB \) is \( 0 \), we need to ensure that \( \frac{2}{18 - 2t^2} \) is undefined or negative, which occurs when \( 18 - 2t^2 \leq 0 \) or \( t^2 \geq 9 \).

### Step 6: Find the maximum area
Substituting \( t^2 = 9 \) into the area formula:
\[ \text{Area} = 2(9) + 36 = 18 + 36 = 54 \]

Thus, the maximum area of the convex polygon is \( \boxed{54} \).