![]() Each phase, launch, staging, orbit insertion, course correction and docking is a piece that has a very different characteristics of fuel consumption, and will require a different expression with different variables (air resistance, weight, gravity, burn rates etc.) at each stage in order to model it correctly. Hmmm, something more scientific? How about modeling the fuel usage of a space shuttle from launch to docking with the ISS. Or perhaps your local video store: rent a game, $5/per game, rent 2-3 games, $4/game, rent more than 5 games, $3/per game.Īsk your folks about tax brackets, another piece-wise function. ![]() In your day to day life, a piece wise function might be found at the local car wash: $5 for a compact, $7.50 for a midsize sedan, $10 for an SUV, $20 for a Hummer. Let the number of regular gadgets manufactured each day = \(x\).Where ever input thresholds (or boundaries) require significant changes in output modeling, you will find piece-wise functions.If a profit of $20 is realized for each regular gadget and $30 for a premium gadget, how many of each should be manufactured to maximize profit? Due to other restrictions, the company can make at most 7 gadgets a day. A regular gadget requires 1 hour of assembly and 2 hours of finishing, while a premium gadget needs 2 hours of assembly and 1 hour of finishing. Each gadget requires the use of two operations, assembly and finishing, and there are at most 12 hours available for each operation. Therefore, we conclude that Niki should work 4 hours at Job I, and 8 hours at Job II.Ī factory manufactures two types of gadgets, regular and premium. Critical PointsĬlearly, the point (4, 8) gives the most profit: $400. To maximize Niki's income, we will substitute these points in the objective function to see which point gives us the highest income per week. ![]() These are found using any methods from Chapter 3 as we are looking for the points where any two of the boundary lines intersect. They are listed as (0, 0), (0, 12), (4, 8), (8, 0). Therefore, we will identify all the vertices (corner points) of the feasible region. The Fundamental Theorem of Linear Programming states that the maximum (or minimum) value of the objective function always takes place at the vertices of the feasible region. The shaded region where all conditions are satisfied is the feasible region or the feasible polygon. Therefore, in this example, we shade the region that is below and to the left of both constraint lines, but also above the x axis and to the right of the y axis, in order to further satisfy the constraints \(x \geq 0\) and \(y \geq 0\). In the graph below, after the lines representing the constraints were graphed, the point (0,0) was used as a test point to determine that ![]()
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