Section 5-2

(10) Maximize:
P = 3x1 + 2x2
  
Subject to...
6x1 + 3x2
≤ 24
3x1 + 6x2
 ≤ 30
x1
≥ 0
x2
≥ 0
[Graphics:Images/5-2_gr_1.gif]

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

 

 [Graphics:Images/5-2_gr_2.gif]

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

 

 [Graphics:Images/5-2_gr_3.gif]

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.