A farmer can plant up to 6 acres of land with soy beans and corn or use of a necessary pesticide is limited by federal regulations to 15 gallons for her entire 6 acres soybean requires 2 gallons of pesticide for every acres planted and corn requires 3 gallons per acre the profit the farmer makes by earning 4000 for every acres of soybeans the plants and 3000 for every acre she plants with barley can be modeled by P = 4000 X +3000 y what is the maximum profit that she can earn

The maximum profit that the farmer can earn given P = 4000x +3000y is 21,000

Given:

P = 4000x +3000y

Let

soyabean = x

corn = y

The equation:

x + y = 6 (1)

2x + 3y = 15 (2)

multiply (1) by 2

2x + 2y = 12 (3)

2x + 3y = 15 (2)

subtract (3) from (2) to eliminate x

3y - 2y = 15 - 12

y = 3

substitute y = 3 into (1)

x + y = 6

x + 3 = 6

x = 6 - 3

x = 3

Therefore, the maximum profit,

P = 4000x +3000y

= 4000(3) + 3000(3)

= 12000 + 9000

= 21,000

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18000

Step-by-step explanation:

X represents acres of soybeans and Y represents acres of corn.  The profit the farmer makes by earning 4000 for every acres of soybeans the plants and 3000 for every acre she plants with corn can be modeled by:

P = 4000X +3000Y        .        .       .    1)

The farmer can plant up to six acres of land. This can be modelled by:

X  +  Y   ≤   6            .            .            .       2)

Necessary pesticide is limited by federal regulations to 15 gallons for her entire 6 acres soybean requires 2 gallons of pesticide for every acres planted and corn requires 3 gallons per acre,

2X   +     3Y  ≤   15         .             .           .   3)

We need to solve equation 2 and 3 simultaneously. Multiplying eqn 2 by 2 to get and subtracting the result from eqn 3 to get:

Y   =   3

Putting Y = 3 in equation 2 we get:

X  = 3

The profit is maximized at X = 3 and Y = 3

Therefore, P = 4000(3) + 2000(3) = 18000

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