Questions: A company manufactures two types of electric hedge trimmers, one of which is cordless. The cord-type trimmer requires 2 hours to make, and the cordless model requires 4 hours. The company has only 800 work hours to use in manufacturing each day, and the packaging department can package only 300 trimmers per day. If the company profits are 36.50 for the cord-type model and 73.00 for the cordless model, how many of each type should the company produce per day to maximize profits? Scenario =1 (with the smaller number of cord-type trimmers): corded models cordless models Scenario =2 (with the larger number of cord-type trimmers): corded models cordless models

A company manufactures two types of electric hedge trimmers, one of which is cordless. The cord-type trimmer requires 2 hours to make, and the cordless model requires 4 hours. The company has only 800 work hours to use in manufacturing each day, and the packaging department can package only 300 trimmers per day. If the company profits are 36.50 for the cord-type model and 73.00 for the cordless model, how many of each type should the company produce per day to maximize profits?

Scenario =1 (with the smaller number of cord-type trimmers):
corded models
cordless models
Scenario =2 (with the larger number of cord-type trimmers):
corded models
cordless models

Transcript text: A company manufactures two types of electric hedge trimmers, one of which is cordless. The cord-type trimmer requires 2 hours to make, and the cordless model requires 4 hours. The company has only 800 work hours to use in manufacturing each day, and the packaging department can package only 300 trimmers per day. If the company profits are $36.50 for the cord-type model and $73.00 for the cordless model, how many of each type should the company produce per day to maximize profits? Scenario $=1$ (with the smaller number of cord-type trimmers): $\square$ corded models $\square$ cordless models Scenario $=2$ (with the larger number of cord-type trimmers): $\square$ corded models $\square$ cordless models

Solution

Solution Steps

To solve this problem, we need to set up a linear programming model. The objective is to maximize the profit function subject to the constraints of manufacturing hours and packaging capacity. Define variables for the number of corded and cordless trimmers. Use these variables to express the constraints and the profit function. Then, use a linear programming solver to find the optimal number of each type of trimmer to produce.

Step 1: Define Variables

Let $ x $ be the number of corded trimmers produced and $ y $ be the number of cordless trimmers produced.

Step 2: Set Up the Objective Function

The profit function to maximize is given by: \[ P = 36.50x + 73.00y \]

Step 3: Establish Constraints

The constraints based on the manufacturing hours and packaging capacity are:

Manufacturing hours: \[ 2x + 4y \leq 800 \]
Packaging capacity: \[ x + y \leq 300 \]
Non-negativity: \[ x \geq 0, \quad y \geq 0 \]

Step 4: Solve the Linear Programming Problem

The solution to the linear programming problem yields: \[ x = 0.0 \quad \text{(cord-type trimmers)} \] \[ y = 200.0 \quad \text{(cordless trimmers)} \]