A farmer has 107 acres on which to plant oats or corn. Each acre of oats requires $18 capital and 2 hours of labor. Each acre of corn requires $36 capital and 6 hours of labor. Labor costs are $8 per hour. The farmer has $2100 available for capital and $2400 available for labor. If the revenue is $47 from each acre of oats and $111

from each acre of corn, what planting combination will produce the greatest total profit? (Profit here is revenue plus leftover capital and labor cash reserve.) What is

the maximum profit?

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Step-by-step explanation:

the objective is to maximize profit

let x be corn and y be oat

let us begging by formulating the linear program for the problem

The objective function is

  111x+47y=P

The constraints are

1. capital

 36x+18y=21001

2. labor

6x(8)+2y(8)=2400

48x+16y=24002

solving 1 and 2 we apply substitution method

36x+18y=21001

36x=2100-18y

x=(2100-18y)/36

put x=(2100-18y)/36 in 2 we have

48(2100-18y)/36+16y=2400

(100800-864y)/36+16y=2400

2800-24y+16y=2400

-8y=2400-2800

-8y=-400

8y=400

y=400/8

y=50

to find x put y=50 in 36x+18y=21001

36x+18(50)=21001

36x+900=2100

36x=2100-900

36x=1200

x=1200/36

x=33.33

profit is

111x+47y=P

111(33.33)+47(50)=P

3699.63+2350=P

P=6049.63

$6049.63

the planning comnination is to work on

33.33 acre of corn and 50 acre of oat

x is variable or an unknown value to find

example:

2x = 5 which means x = 3

the image of (6, 9) under a dilation is (4, 6). the scale factor is