Maximum and Minimum

2) Find the maximum value of P = x + 4y subject to the constraints
y–x≤0
x≤4
y≥0
x + 2y ≤ 6
4) Find the minimum value of C = 2x + 3y subject to the constraints
2x + y ≥ 18
x + y ≥ 12
3x + 2y ≤ 34
6) Find the maximum value of P = 2x + 3y subject to the constraints
y–x≤2
x+y≤4
0≤x≤3
y≥0
12) A nutritionist is designing a meal for one of her patients. The meal must include two
vegetables, A and B, but not more than 100 g of each. Suppose that each 10-g portion of A
contains 2 units of iron and 2 units of vitamin B12, and each 10-g portion of B contains 1 unit of
iron and 5 units of vitamin B12. The number of calories in each 10-g portion of these vegetables
is 5 for A and 3 for B. If the patient needs at least 20 units of iron and 36 units of vitamin B12 in
the meal, how many grams of each vegetable should the nutritionist include to satisfy the iron
and vitamin requirements while minimizing the number of calories in the meal?
16) A local television station is faced with a problem. It found that program A with 20 min of
music and 1 min of commercials draws 30,000 viewers, whereas program B with 10 min of
music and 1 min of commercials draws 10,000 viewers. The sponsor insists that at least 6 min
per week be devoted to his commercials, and the station can afford no more than 80 min of music
per week. How many times per week should each program be run to obtain the maximum
number of viewers?
20) The Excelsior Mining Company operates two mines, EMC 1 and EMC 2. EMC 1 produces
20 tons of lead ore and 30 tons of low grade silver ore per day of operation. EMC 2 produces 15
tons of lead ore and 35 tons of low-grade silver ore per day of operation. Lead ore sells for $14
per ton and low-grade silver ore sells for $34 per ton. The company can sell at most 630 tons of
the low-grade silver ore per month, but it can sell all the lead ore it produces. However, there is
no space available for stockpiling any silver ore. The company employs one crew and operates
only one of the mines at a time. Furthermore, union regulations stipulate that the crew not be
worked in excess of 20 days per month. How many days per month should Excelsior schedule
for each mine so that the income from the sale of the ore is at a maximum?