Chapter 11, Problem 102RE

To determine

How many acres should be allocated for each crop?

Answer to Problem 102RE

  $\boxed{466.25acres}$

  $\boxed{533.75acres}$

Explanation of Solution

Given information:

A $1000-acre$ farm in millions is used to grow corn and soybeans. The cost per acre for raising corn is $\$65$ , and the cost per acre for soybean is $\$45$ . If $\$54,325$ has been budgeted for costs and all the acreage is to be used, how many acres should be allocated for each crop?Calculation:

Let the acres of planted corn and soya bean in millions be $x$ and $y$ respectively.

Total area of the land is $1000-acre$ be written as,

  $x+y=1000$ ..... $\left(1\right)$

The cost per acre for raising corn is $\$65$ and soybean is $\$45$ .the total budget of the farmer is $\$54,325$

  $\begin{array}{l}65x+45y=54,325\\ 13x+9y=\mathrm{10865....}\left(2\right)\end{array}$

Now solve both equations by elimination method by multiplying equation $\left(1\right)$ by $13$ and equation $\left(2\right)$ by $1$ .and subtract equation $\left(2\right)$ by equation $\left(1\right)$

  $\begin{array}{l}\underset{\_}{\begin{array}{l}13x+13y=13000\\ 13x+9y=10865\end{array}}\\ 4y=2135\\ y=2135/4=533.75\end{array}$

Now substitute $y=533.75$ in equation $\left(1\right)$ . We get,

  $\begin{array}{l}x+y=1000\\ x+533.75=1000\\ x=1000-533.75\\ x=466.25\end{array}$

Hence, the acres should be allocated for corn is $\boxed{466.25acres}$ and soybean $\boxed{533.75acres}$ .