Constraints Objective Function Vertices x y Analysis Work

_____________ Constraints: Objective Function: Vertices (x, y) Analysis: Work - plug into p(x, y) Solution

Boots Unplugged: Mrs. Boots runs a radio show. She gets paid $3 for each caller (x) and $5 for each advertisement (y). She cannot have a negative number of callers, however the number of callers must be greater than the number of advertisements minus one. She cannot have a negative number of advertisements, however they must be less than -1/4 times the number of callers plus six. Constraints: Number of advertisements Objective Function: Paid is p(x, y) = 3 x + 5 y Vertices (x, y) Work - plug into p(x, y) Solution (0, 0) 3(0) + 5(0) 0 (4, 5) 3(4) + 5(5) 37 (25, 0) 3(25) + 5(0) 75 Analysis: The maximum amount paid occurs when there are 25 callers & no advertisements. (4, 5) (0, 1) (25, 0) Number of callers

Steps 1. 2. 3. 4. 5. 6. 7. 8. Write your problem statement containing limits. Write an inequality for each constraint. Determine the quantity to be maximized or minimized and write the P(x, y) function. This is called the objective function. Label each axis and graph each function. All four constraint equations are graphed on the same coordinate plane. Remember the graphing process is the same as graphing a line in y = mx + b form, only < or > has a dashed line. For shading, < is down (or left) and > is up (or right). The feasible region is the area shaded by all four constraints. Darken this area which represents the system of possible answers. Identify the vertices or corners of the feasible region. Write these points (where the constraints intersect) in the table. Evaluate the objective function for each vertex. From the results determine the maximum or minimum and the conditions by which it occurs. Write a statement containing your answer.

Smart. Cars A company manufactures bolts for Fords & bolts for Smart. Cars. One machine can produce no more than 70 Ford fender bolts daily and another machine can produce a maximum of 50 Smart. Car bumper bolts. The combined number of fender bolts and bumper bolts that the packaging department can handle is 90 daily. How many of each type of bolts should the company manufacture for maximum daily income if the Smart. Car bumper bolts sell for $6 and the Ford fender bolts sell for $4? What is the maximum daily income? Graph constraints to find the corners Plug corners (x, y) in this equation to find profit.

Wheel Cycle Co. The Wheel Cycle Co. manufactures motorcycles & bicycles. To stay in business it must produce at least 10 motorcycles each month, but it does not have the facilities to produce more than 60. It also does not have the facilities to produce more than 120 bicycles. The total production cannot exceed 160. The profit on a motorcycle is $134 & on a bicycle $20. Find the number of each that must be produced to achieve a maximum profit. Graph constraints to find the corners Plug corners (x, y) in this equation to find profit.

Carpenter A carpenter makes bookcases in two sizes, large & small. It takes about 6 hours to make a large bookcase & 2 hours to make a small one. The profit on a large bookcase is $50 and the profit on a small one is $20. The carpenter can only spend 24 hours a week making bookcases and must make at least 2 of each size. How many of each must be made per week to achieve a maximum profit? What will the profit be? Graph constraints to find the corners Plug corners (x, y) in this equation to find profit.

Salon The Cutting Edge Salon schedules appointments for 30 minutes haircuts and 1 hour for highlights. Each haircut costs $20 & highlights cost $45. The salon wants to schedule no more than 4 highlights per day and at least 3 haircuts. Find the number of appointments that produces the maximum income for a stylist in an 8 hour workday. Graph constraints to find the corners Plug corners (x, y) in this equation to find profit.