Section 5-2
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First, we plot the feasible region (above) and note that the two lines intersect at (2, 4). The region is bounded, so Theorem 2 says an optimal solution will exist. Theorem 1 says the optimal solution will happen at a corner point. Therefore we evaluate the objective function at each corner point:
Corner points | P = 3x1 + 2x2 |
(0, 0) | 3(0) + 2(0) = 0 |
(0, 5) | 3(0) + 2(5) = 10 |
(2, 4) | 3(2) + 2(4) = 14 |
(4, 0) | 3(4) + 2(0) = 12 |
From the table, we see that the optimal solution occurs when x1 = 2, and x2 = 4
(32-A) Maximize profit given the following data.
Table | Chair | max hours per day | |
Assembly | 8 hours | 2 hours | 400 hours |
Finishing | 2 hours | 1hour | 120 hours |
Profit | $90 | $25 |
Let x be the number of tables produced.
Let y be the number of chairs produced.
The profit is P = 90x + 25y.
The assembly time is 8x + 2y hours, and the finishing time is 2x + y hours.
Thus we wish to
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The feasible region is graphed above. It is bounded, so the optimal solution exists and occurs at a corner point. The corner points are obtained and plugged into the profit function:
Corner points | Profit P = 90x + 25y |
(0, 0) | 90(0) + 25(0) = $0 |
(0, 120) | 90(0) + 25(120) = $3000 |
(50, 0) | 90(50) + 25(0) = $4500 |
(40, 40) | 90(40) + 25(40) = $4600 |
So you can see that the maximum profit happens when 40 chairs and 40 tables are produced.
(42) Start by putting the information into a table.
Food M | Food N | min daily requirement | |
calcium | 30 units | 10 units | 360 units |
iron | 10 units | 10 units | 160 units |
vitamin A | 10 units | 30 units | 240 units |
Cholesterol | 8 units | 4 units |
Let x be the number of ounces of Food M.
Let y be the number of ounces of Food N.
Then the total cholesterol is C = 8x + 4y units.
The total calcium is 30x +10y units.
The total iron is 10x + 10y units.
The total vitamin A is 10x + 30y units.
So we want to...
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The feasible region is graphed above. Find the corner points. Plug them into the Cholesterol formula.
Point | Cholesterol C = 8x +4y |
(0, 36) | 8(0) + 4 (36) = 144 units |
(24, 0) | 8(2) + 4 (0) = 192 units |
(10, 6) | 8(10) + 4 (6) = 104 units |
(12, 4) | 8(12) + 4 (4) = 112 units |
You can see that the cholesterol is minimized if you have 10 ounces of Food M, and 6 ounces of Food N.