Section 52

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 = 3x_{1} + 2x_{2} 
(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 x_{1} = 2, and x_{2} = 4
(32A) 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

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...

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.