A manufacturer makes two kinds of computers, laptops and desktops. The company has equipment to manufacture any number of laptops up to 600 per month or any number of desktops up to 800 per month. However, the company can manufacture at most 1,000 computers of both kinds per month in all. It takes 35 man-hours to manufacture either type of computer and the manufacturer has available 25,000 man-hours per month. The manufacturer earns a profit of \( \$ 175 \) on each laptop and \( \$ 100 \) on each desktop. Write an objective function to model this situation if the manufacturer wants to maximize his profit, using x for the number of laptops and y for the number of desktops.

To model the situation described, we need to define the variables and the objective function based on the information provided. ### Step 1: Define the Variables Let: - \( x \) = number of laptops produced per month - \( y \) = number of desktops produced per month ### Step 2: Identify the Profit Function The profit earned from each type of computer is given as follows: - Profit from each laptop = \( 175 \) - Profit from each desktop = \( 100 \) Thus, the total profit \( P \) can be expressed as: \[ P = 175x + 100y \] ### Step 3: Write the Objective Function The objective function, which we want to maximize, is: \[ \text{Maximize } P = 175x + 100y \] ### Step 4: Identify the Constraints From the problem, we have the following constraints: 1. **Production Capacity Constraints:** - The maximum number of laptops: \( x \leq 600 \) - The maximum number of desktops: \( y \leq 800 \) 2. **Total Production Constraint:** - The total number of computers (laptops and desktops) cannot exceed 1,000: \[ x + y \leq 1000 \] 3. **Man-Hours Constraint:** - Each computer (laptop or desktop) takes 35 man-hours, and the total available man-hours are 25,000: \[ 35x + 35y \leq 25000 \] This can be simplified to: \[ x + y \leq 714.29 \quad (\text{approximately } 714) \] ### Summary The objective function and constraints can be summarized as follows: **Objective Function:** \[ \text{Maximize } P = 175x + 100y \] **Subject to Constraints:** \[ \begin{align*} x & \leq 600 \\ y & \leq 800 \\ x + y & \leq 1000 \\ x + y & \leq 714 \\ x, y & \geq 0 \end{align*} \] This formulation allows the manufacturer to maximize profit while adhering to production and resource constraints.