Find the maximum and minimum values of each objective function ov...

Question

Find the maximum and minimum values of each objective function over the region of feasible solutions shown at the right.  objective function = 3x + 5y

Accepted Answer

Welcome back. I am so glad you're here. We are asked to evaluate the objective function Z at the vertices of the graph of the region of feasible solutions and determine the maximum and minimum values of the function. Our function is Z equals five X plus eight Y. And then we're given a graph, our graph has a shape with vertices at 29, 10, 11 10, 1 and 43. And our answer choices are answer choice. A, a maximum of 138. A minimum of 44 answer choice B A maximum of 138 a minimum of answer choice C A maximum of 58 a minimum of 44 and answer choice D A maximum of 116 a minimum of 48. All right. So when we are evaluating an objective function, we take each of our vertices so we can start with 29. And we are going to plug the X values in for X in our objective function and the Y values in for Y in our objective function. We recall from previous lessons that in an ordered pair, we list the X values first followed by the Y values. So for the, for the vertex 29 will take five multiplied not by X but by the X value here of two plus eight multiplied by the Y value of nine. And that's going to simplify to be 10 plus 72 which equals 82. Now, for our next vertex at 10 11, we will plug in for the X value 10. So we have five multiplied by 10 plus eight, multiplied by the Y value which is 11 five, multiplied by 10 is 50 plus eight, multiplied by 11 is 88. 50 plus 88 is 138. Now, for the next vertex at 10, 1 plugging in 10, for the X value five multiplied by 10 plus eight multiplied by the Y value which is one five multiplied by 10 is 50 plus eight, multiplied by one is eight. So this simplifies to 58. Now for our final vertex, 43 plugging in the X value four, so five multiplied by four plus eight, multiplied by the Y value three, five multiplied by four is 20 plus eight, multiplied by three, is 24. 20 plus 24 is 44. Now, we want to find the highest of those four values and that is 138 138 is our maximum. Now, we find the lowest of those values that's 44. So 44 is our minimum. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Find the maximum and minimum values of each objective function over the region of feasible solutions shown at the right. objective function = 3x + 5y

Key Concepts

Objective Function

Feasible Region

Linear Programming

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