Section 5-2

 (10) Maximize: P = 3x1 + 2x2 Subject to... 6x1 + 3x2 ≤ 24 3x1 + 6x2 ≤ 30 x1 ≥ 0 x2 ≥ 0

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

 maximize P = 90x + 25y subject to... 8x + 2y ≤ 400 2x +y ≤ 120 x ≥ 0 y ≥ 0

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...

 minimize C = 8x +4y ...subject to 30x +10y ≥ 360 10x + 10y ≥ 160 10x + 30y ≥ 240 x ≥ 0 y ≥ 0

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.