Question

How many acres of land will be left unplanted if the farmer aims for a maximum profit?

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

let $x$ be the acres of wheat planted and

$y$ be the acres of rye planted

Given that there are a total of $10$ acres of land to plant.

Atleast $7$ acres is to be planted i.e., $x+y\ge 7$

Given that the cost to plant one acre of wheat is $\$200$

Therefore, the cost for $x$ acres of wheat is $200x$

Given that the cost to plant one acre of rye is $\$100$

Therefore, the cost for $y$ acres of rye is $100y$

Given that, an amount for planting wheat and rye is $\$1200$

Therefore the total cost to plant wheat and rye is $200x+100y\le 1200⟹2x+y\le 12$

Given that, the time taken to plant one acre of wheat is $1$ hr

Therefore, the time taken to plant $x$ acres of wheat is $x$ hrs

Given that, the time taken to plant one acre of rye is $2$ hrs

Therefore, the time taken to plant $y$ acres of rye is $2y$ hrs

Given that, the total time for planting is $12$ hrs

Therefore, the total time to plant wheat and rye is $x+2y\le 12$

Given that, one acre of wheat yields a profit of $\$500$

Therefore, the profit from $x$ acres of wheat is $500x$

Given that, one acre of rye yields a profit of $\$300$

Therefore, the profit from $y$ acres of wheat is $300y$

therefore the total profit from the wheat and rye is $P=500x+300y$

In the above figure, the blue shaded region is the feasible region with three corner points.$(4,4),(2,5),(5,2)$

Now substituting the corner points the profit equation,

substituting $(4,4)⟹P=500x+300y=500\left(4\right)+300\left(4\right)=3200$

substituting $(2,5)⟹P=500x+300y=500\left(2\right)+300\left(5\right)=2500$

substituting $(5,2)⟹P=500x+300y=500\left(5\right)+300\left(2\right)=3100$

$\$3200$ is the maximum profit is attained by planting $4$ acres of wheat and $4$ acres of rye.

i.e., a total of $4+4=8$ acres of land is planted. But, the total land is $10$ acres. therefore, the unplanted land is $10-8=2$ acres