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Why do we test only intersection points? Does this always work?

With GeoGebra we can easily test all of our points and see when and why we need test only intersection points.

Example with YouTube Mathcast and GeoGebra InterActivity

Let's look again at Example 1 - except we will pretend that x and y can be decimals.

Our constraints were: x ≥0 andy ≥0,10x+20y≤140 and 0.6x+0.8y≤7.2 and our objective was to maximize volume: V=0.4x+0.6y.

Example 1:We need to put file cabinets in our office. File cabinet A costs 10 Euros, has a footprint of 0.6m^{2}, and a storage capacity of 0.4m^{3}. File cabinet B costs 20 Euros, has a footprint of 0.8m^{2}, and a capacity of 0.6m^{3}. We can spend a maximum of 140 Euros and we have a maximum floor space of 7.2 m^{2}. How many cabinets of type A and type B should we purchase to maximize our storage capacity? Our solution was x=8 and y=3 and it gave a maximum volume V=5.

Question:What points x and y satisfy our constraints and give a volume V=2? How about a volume of V=3 or V=5 or V=6?

YouTube Mathcast

Exploring InterActivity with GeoGebra Directions for InterActivity

Remember - only points in the purple-shaded area satisfy our constraints.
• Click & drag the slider to the desired V=Volume. When substituted into the objective function all points on dashed line make this Volume.
• To see what point (x,y) fit our constraints and make this Volume, click & drag the point on the dashed line within the purple-shaded area.
• Click & drag Volume=5. Notice that the only point in the purple-shaded area with V=5 is (x,y)=(8,3).
• Click & drag Volume>5. Now, the dashed line does not intersect the purple-shaded area! What does this mean?

DIY1 - You make a slider for the objective function of Example 1.

Watch the YouTube mathcast and then DIY in the GeoGebra interactivity below.

Example with YouTube Mathcast and GeoGebra InterActivity

YouTube Mathcast

DIY InterActivity with GeoGebra Directions for InterActivity

Remember - only points in the purple-shaded area satisfy our constraints.
• Check the range of values of the objective function. Here they are 0-5 so we will make slider 0-10.
• Click on the slider tool and then anywhere in Drawing pad. Change name to Volume , min=0, max=10, increment=0.1.
• Click in the Input bar (bottom left) and type the objective function: Volume=0.6x+0.8y and hit Enter.

• Click on new point tool and click on the new line.
• Decorate slider, line and point as desired.

DIY2 - You make a slider for the objective function of Example 3.

Watch the YouTube mathcast and then DIY in the GeoGebra interactivity below.

Example with YouTube Mathcast and GeoGebra InterActivity

YouTube Mathcast

DIY InterActivity with GeoGebra Directions for InterActivity

Remember - only points in the dark triangle satisfy our constraints.
• Check the range of values of the objective function. Here they are 1550-2548 so we will make slider 1000-3000.
• Click on the slider tool and then anywhere in Drawing pad. Change name to Calories , min=1000, max=3000, increment=1.
• Click in the Input bar (bottom left) and type the objective function: Calories=100x+115y and hit Enter.

• Click on new point tool and click on the new line.
• Decorate slider, line and point as desired.

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