Chapter 2.7, Problem 14E

(a)

To determine

To find: The number of acres of each should he plant to earn the greatest profit.

Answer to Problem 14E

The farmer will use120 acres to be planted with corn and 60 acres to be planted with soybeans.

Explanation of Solution

Given:

An American farmer has 180 acres on which to grow corn and soybeans. He is planting a least 40 acres of corn and 20 acres of soybeans.

If the farmers plants at least 2 acres of corn for every acre of soybeans.

Let $x$ be the number of acres to be planted with corn.

And let $y$ be the number of acres to be planted with soybeans.

The profit function $P\left(x,y\right)$ is given by as below:

  $P\left(x,y\right)=150x+250y$ .

The following inequalities must be simultaneously satisfied:

  $\begin{array}{c}x\ge 40\\ y\ge 20\\ x+y\le 180\\ x\ge 2y\left[\text{given}\text{farmers plants at least 2 acres of corn for every acre of soybeans}.\right]\end{array}$

The graph of the above inequalities is given below:

  

The feasible region is the triangle with vertices $\left(40,20\right),\left(20,60\right),\text{and}\left(160,20\right)$

A theorem says that the extreme values of $P$ occurs at one or more of these vertices, so evaluate $P$ at each vertices.

  $\begin{array}{c}P\left(40,20\right)=150\left(40\right)+250\left(20\right)\\ =11000\\ P\left(120,60\right)=150\left(120\right)+250\left(60\right)\\ =33000\end{array}$

Further simplified as:

  $\begin{array}{c}P\left(160,20\right)=150\left(40\right)+250\left(20\right)\\ =29000\end{array}$

Therefore the farmer will use120 acres to be planted with corn and 60 acres to be planted with soybeans.

(b)

To determine

To find: The farmer’s maximum profit.

Answer to Problem 14E

The maximum profit of farmer will be $\$33000$ .

Explanation of Solution

It is found that $P\left(120,60\right)=33000$ is the maximum of the function.

So the maximum profit of farmer will be $\$33000$ .

Therefore, the maximum profit of farmer will be $\$33000$ .