Work with a partner.Linear programming is a modeling technique that is useful for quantitative decisions in engineering, business, and the sciences. To solve a linear programming problem, you must find the maximum and minimum values of a linear equation within a set of constraints expressed as inequalities. This is called the feasible region, or the solution set of a system of linear inequalities. The extreme values, or maximum and minimum values, of any objective function f (x, y) must occur at the vertices of the feasible region
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Question 1
1.
Use technology to graph the system of inequalities.
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Question 2
2.
Complete the table by evaluating the function f (x, y) = x + 3y at the vertices of the feasible region.
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Question 3
3.
The vertex of f that has a minimum value is (_______).
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Question 4
4.
The vertex of f that has a maximum value is (_______).
By evaluating the objective function for several other points in the feasible region, you can see that the value of the objective function increases without bound as the coordinates increase. So, the objective function has no maximum value on the feasible region. The minimum value is 5. It occurs when x = 0 and y = 5.
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Question 5
5.
Find the minimum value and the maximum value of the objective function subject to the given constraints.
The minimum value is _______ at the coordinate (_______).
The maximum value is _______ at the coordinate (_______).
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Question 6
6.
Find the minimum value and the maximum value of the objective function subject to the given constraints.
The minimum value is _______ at the coordinate (_______).
The maximum value is _______ at the coordinate (_______).
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Question 7
7.
Find the minimum value and the maximum value of the objective function subject to the given constraints.
The minimum value is _______ at the coordinate (_______).
The maximum value is _______ at the coordinate (_______).
Real-Life Application
Two woodshops make the same kind of garden shed. The table gives the preparation and assembly times (in hours) required to make one shed in each shop. For the two shops combined, the manufacturer can afford to use up to 200 hours for preparation and up to 240 hours for assembly per week. Shop A earns a profit of $300 per shed and Shop B earns a profit of $400 per shed. How many sheds per week should the manufacturer make in each shop to maximize profit?
A maximum profit of $8800 is obtained by making 8 sheds in Shop A and 16 sheds in Shop B.