Questions: Objective Function for Maximum Profit: z=19x+25y Constraint inequality for cutting and sanding: 45x+55y ≤ 300; Constraint inequality for sealing and attaching handles: 15x+25y ≤ 180

Solution Steps

To solve this problem, we need to determine the correct objective function and constraints for maximizing the artist's profit. The objective function is based on the profit from each type of tray, and the constraints are based on the time limitations for cutting, sanding, sealing, and attaching handles. We will use the given information to set up the objective function and inequalities.

1. Define the objective function for profit based on the number of trays and their respective profits.
2. Set up the constraint inequalities for the time spent on cutting and sanding, and sealing and attaching handles, based on the given time limits.

The objective function for maximizing the artist's daily profit is given by: \[ z = 19x + 25y \] where \( x \) is the number of 12-inch trays and \( y \) is the number of 18-inch trays.

The constraints based on the time limitations are:

1. Cutting and sanding constraint: \[ 45x + 55y \leq 300 \]
2. Sealing and attaching handles constraint: \[ 15x + 25y \leq 180 \]

Using the constraints and the objective function, we solve the linear programming problem to find the values of \( x \) and \( y \) that maximize \( z \).

The solution to the linear programming problem gives a maximum profit of approximately \( 136.36 \).

Final Answer