1. A farmer has 100 acres of land on which she plans to grow wheat and corn. Each acre of wheat requires 4 hours of labor and $20 of capital, and each acre of corn requires 16 hours of labor and $40 of capital. The farmer has at most 800 hours of labor and $2400 of capital available. If the profit from an acre of wheat is $80 and from an acre of corn is $100, how many acres of each crop should she plant to maximize her profit?
c = acres of corn
w = acres of wheat
constraints:
c+w </= 100
graph that line,shade under
wheat cost = 20 w
corn cost = 40 c
20 w + 40 c </= 2400
graph that line, shade under
wheat labor = 4 w
corn labor = 16 c
4 w + 16 c </= 800
graph that line, shade under
Now the criterion
p = 80 w + 100 c
check p at every intersection of the above constraint lines, pick the biggest p
Corn: 0 , Wheat: 120
Let X be the area for wheat (in acres), and
Let Y be the area for corn.
The amount of labor is 5X + 15Y hours.
The amount of capital is 20X + 30Y dollars