`Z = 15x + 30y`
`x + y le 8, 2x + y ge 8, x - 2y ge 0, x, y ge 0`

Answer

To solve the linear programming problem given by the objective function \( Z = 15x + 30y \) with the constraints:

1. \( x + y \leq 8 \)
2. \( 2x + y \geq 8 \)
3. \( x - 2y \geq 0 \)
4. \( x \geq 0 \)
5. \( y \geq 0 \)

we will follow these steps:

### Step 1: Write the objective function and constraints
The objective function is:
\[ Z = 15x + 30y \]

The constraints are:
1. \( x + y \leq 8 \)
2. \( 2x + y \geq 8 \)
3. \( x - 2y \geq 0 \)
4. \( x \geq 0 \)
5. \( y \geq 0 \)