Questions: The graph of the feasible region is shown. f=8x+7y Find the corners of the feasible region. (Order your answers from smallest to largest x, then from smallest to largest y.) (x, y)= (x, y)= (x, y)= (x, y)= (x, y)= Find the maximum and minimum of the given objective function (if they exist). (If an answer does not exist, enter DNE.) maximum f= minimum f=

The graph of the feasible region is shown. f=8x+7y Find the corners of the feasible region. (Order your answers from smallest to largest x, then from smallest to largest y.) (x, y)= (x, y)= (x, y)= (x, y)= (x, y)= Find the maximum and minimum of the given objective function (if they exist). (If an answer does not exist, enter DNE.) maximum f= minimum f=
Transcript text: The graph of the feasible region is shown. \[ f=8 x+7 y \] Find the corners of the feasible region. (Order your answers from smallest to largest $x$, then from smallest to largest $y$.) \[ \begin{array}{l} (x, y)=(\square) \\ (x, y)=(\square) \\ (x, y)=(\square) \\ (x, y)=(\square) \\ (x, y)=(\square) \end{array} \] Find the maximum and minimum of the given objective function (if they exist). (If an answer does not exist, enter DNE.) maximum $f=$ $\square$ minimum $\quad f=$ $\square$
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Solution

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Solution Steps

Step 1: Find the intersection of 2x + y = 40 and 8x + 5y = 170

Multiply the first equation by 5: 10x + 5y = 200. Subtract the second equation from this result: (10x + 5y) - (8x + 5y) = 200 - 170. Simplify to get 2x = 30, so x = 15. Substitute x = 15 into 2x + y = 40: 2(15) + y = 40, which simplifies to 30 + y = 40, so y = 10. The intersection point is (15, 10).

Step 2: Find the intersection of 2x + y = 40 and x + 5y = 100

Multiply the first equation by 5: 10x + 5y = 200. Subtract the second equation: (10x + 5y) - (x + 5y) = 200 - 100. Simplify to get 9x = 100, so x = 100/9. Substitute x = 100/9 into 2x + y = 40: 2(100/9) + y = 40, which simplifies to 200/9 + y = 40. So y = 40 - 200/9 = (360 - 200)/9 = 160/9. The intersection point is (100/9, 160/9).

Step 3: Find the intersection of x + 5y = 100 and 8x + 5y = 170

Subtract the first equation from the second: (8x + 5y) - (x + 5y) = 170 - 100. Simplify to get 7x = 70, so x = 10. Substitute x = 10 into x + 5y = 100: 10 + 5y = 100, which simplifies to 5y = 90, so y = 18. The intersection point is (10, 18).

Final Answer:

The corners, ordered as requested, are: (10, 18), (15, 10), (100/9, 160/9). The objective function is maximized at (10,18) where f = 8(10) + 7(18) = 80 + 126 = 206, and minimized at (15,10) where f = 8(15) + 7(10) = 120 + 70 = 190.

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