To solve this problem, we can use the method of linear programming. Linear programming is a mathematical technique used to determine the optimal solution to a problem given a set of linear constraints. In this case, we want to maximize the objective function Z = 100x + 120y, subject to the constraints:

1) 2x + 3y ≤ 30
2) 3x + y ≤ 17
3) x ≥ 0
4) y ≥ 0

To solve this problem, we can follow these steps:

Step 1: Plot the feasible region
- To visualize the feasible region, we need to graph the inequalities.
- Start by graphing the lines 2x + 3y = 30 and 3x + y = 17.
- To do this, we can find the intercepts and plot the lines.
- For the first line, when x = 0, y = 10 and when y = 0, x = 15.
- For the second line, when x = 0, y = 17 and when y = 0, x = 5.67 (approximately).
- Plot these points and draw the lines.

Step 2: Identify the feasible region
- The feasible region is the area of the graph that satisfies all the constraints.
- Shade the region that is below both lines and in the positive quadrant (x ≥ 0, y ≥ 0).

Step 3: Identify the corner points
- The optimal solution lies at one of the corner points of the feasible region.
- Identify the coordinates of the corner points by finding the intersection of the lines.
- In this case, the corner points are (0, 0), (5.67, 0), and (0, 10).

Step 4: Evaluate the objective function at each corner point
- Substitute the coordinates of each corner point into the objective function Z = 100x + 120y.
- Calculate the value of Z for each point.
- Z(0, 0) = 100(0) + 120(0) = 0
- Z(5.67, 0) = 100(5.67) + 120(0) = 567
- Z(0, 10) = 100(0) + 120(10) = 1200

Step 5: Determine the maximum value of Z
- Compare the values of Z at each corner point and identify the maximum value.
- In this case, the maximum value of Z is 1200 at the point (0, 10).

Therefore, the correct answer is option A) 1200.

1260